Worst Exponential Decay Rate for Degenerate Gradient flows subject to persistent excitation
aa r X i v : . [ m a t h . O C ] J un WORST EXPONENTIAL DECAY RATE FOR DEGENERATEGRADIENT FLOWS SUBJECT TO PERSISTENT EXCITATION ∗ YACINE CHITOUR † , PAOLO MASON † , AND
DARIO PRANDI † Abstract.
In this paper we estimate the worst rate of exponential decay of degenerate gradientflows ˙ x = − Sx , issued from adaptive control theory [3]. Under persistent excitation assumptions onthe positive semi-definite matrix S , we provide upper bounds for this rate of decay consistent withpreviously known lower bounds and analogous stability results for more general classes of persistentlyexcited signals. The strategy of proof consists in relating the worst decay rate to optimal controlquestions and studying in details their solutions.As a byproduct of our analysis, we also obtain estimates for the worst L -gain of the time-varyinglinear control systems ˙ x = − cc ⊤ x + u , where the signal c is persistently excited , thus solving an openproblem posed by A. Rantzer in 1999, cf. [14, Problem 36].
1. Introduction.
The focus of this paper is the convergence rate to the originassociated with descent algorithms of the form(DGF) ˙ x ( t ) = − S ( t ) x ( t ) , x ∈ R n , where S is a locally integrable positive semi-definite n × n symmetric matrix. When-ever S is not positive definite, these dynamics are usually referred to as degenerategradient flow systems . They appear in the context of adaptive control and identifica-tion of parameters (cf. [2, 4, 8, 15]). Of particular importance among the dynamics(DGF), is the case where the rank of S ( t ) is assumed to be at most one, i.e., S = cc ⊤ with c ∈ R n .In order to guarantee global exponential stability (GAS) of (DGF), we assume S to satisfy the persistent excitation condition. That is, there exists a, b, T > a Id n ≤ Z T + tt S ( τ ) dτ ≤ b Id n , ∀ t ≥ . Here, Id n ∈ R n × n is the identity matrix, and the inequalities are to be understood inthe sense of symmetric forms. Clearly, this condition is invariant under conjugation byorthogonal matrices and is actually equivalent to uniform global exponential stabilityof (DGF), cf. [2]. Note also that Condition (PE) has been considered in stabilizationissues for linear control systems with unstable uncontrolled dynamics, cf. [9, 10].Our purpose is to study the worst exponential decay rate R ( a, b, T, n ) of persis-tently excited signals, as a function of the parameters a, b, T > n ∈ N . Letting Sym ( P E ) n ( a, b, T ) denote the family of signals satisfying (PE), this isdefined by(1.1) R ( a, b, T, n ) = inf n R ( S ) | S ∈ Sym ( P E ) n ( a, b, T ) o , where R ( S ) is the exponential decay rate of (DGF), given in terms of the fundamentalmatrix Φ S ( t,
0) of (DGF) by(1.2) R ( S ) := − lim sup t → + ∞ log k Φ S ( t, k t . ∗ This research was partially supported by the iCODE Institute, research project of the IDEXParis-Saclay, and by the Hadamard Mathematics LabEx (LMH) through the grant number ANR-11-LABX-0056-LMH in the “Programme des Investissements d’Avenir”. † Universit Paris-Saclay, CNRS, CentraleSuplec, Laboratoire des signaux et systmes, 91190, Gif-sur-Yvette, France. Email: { yacine.chitour, paolo.mason, dario.prandi } @centralesupelec.fr he literature on the worst decay rate is extensive (cf., e.g., [2, 4, 8, 16]), butmostly restricted to lower bounds. In our context, these results boils down to theexistence of an universal constant C > R ( a, b, T, n ) ≥ Ca (1 + nb ) T .
Our main result is the following, which shows the optimality of this lower bound, for n fixed. Theorem
There exists C > such that, for every < a ≤ b , T > andinteger n ≥ , the worst rate of exponential decay R ( a, b, T, n ) defined in (1.1) satisfies (1.4) R ( a, b, T, n ) ≤ C a (1 + b ) T .
Moreover, the same result holds true when restricting (1.1) to matrices S verifying (PE) with rank at most (i.e., S = cc ⊤ and c ∈ R n ).Remark R ( a, b, T, n ) tends to zero as b tends to infinity. This is in accordance with [7], where it is proved that in general thereis no convergence to the origin for trajectories of (DGF) if only the left inequality of(PE) holds true, i.e., b = + ∞ . More precisely, the authors put forward a “freezing”phenomenon by showing that in this case there exist trajectories of (DGF) whichconverge, as t tends to infinity, to points different from the origin. L -gain of degenerate flows with linear inputs. As a consequence ofTheorem 1.1 and of the arguments to derive it, we solve the first part of a problemby A. Rantzer [14, Problem 36], that we now present. Consider the control system(1.5) ˙ x ( t ) = − c ( t ) c ( t ) ⊤ x ( t ) + u ( t ) , where x, c, u take values in R n . For u ∈ L ([0 , ∞ ) , R n ), let x u ∈ L ([0 , ∞ ) , R n ) bethe trajectory of (1.5) associated with u and starting at the origin. Whenever cc ⊤ satisfies (PE), the trajectories of the uncontrolled dynamics tend to zero exponen-tially, so that the input/output map u x u is well-defined as a linear operator on L ([0 , ∞ ) , R n ) and its L -gain γ ( c ) = sup = u ∈ L ([0 , ∞ ) , R n ) k x u k k u k is finite. (Here, k · k stands for the norm in L ([0 , ∞ ) , R n ).) Rantzer’s question consists in estimating(1.6) γ ( a, b, T, n ) := sup (cid:8) γ ( c ) | cc ⊤ satisfy (PE) (cid:9) . In that direction, we obtain the following result.
Theorem
There exists c , c > such that, for every < a ≤ b , T > andinteger n ≥ , (1.7) c T (1 + b ) a ≤ γ ( a, b, n, T ) ≤ c T (1 + nb ) a . Remark cc ⊤ in (1.5) by a positivesemi-definite n × n symmetric matrix satisfying (PE). Recently, there has been an increas-ing interest in considering more general types of persistent excitation conditions,cf. [6, 13, 11]. We focus on the following generalized persistent excitation condition:(GPE) a ℓ Id n ≤ Z τ ℓ +1 τ ℓ S ( t ) dt ≤ b ℓ Id n , here ( a ℓ ) ℓ ∈ N , ( b ℓ ) ℓ ∈ N are sequences of positive numbers, and ( τ ℓ ) ℓ ∈ N is a strictlyincreasing sequence of positive times such that τ ℓ → + ∞ as ℓ → + ∞ .An important question consists in determine under which condition (GPE) guar-antees global asymptotic stability (GAS) for (DGF). The following sufficient conditionis known:(1.8) ∞ X ℓ =0 a ℓ b ℓ = + ∞ . This has been proved in [13] (cf. also [6]) for the case where S has rank at most one.The same argument can be extended to the general case, cf., [8].As a byproduct of our analysis, we show that this condition is indeed necessary. Theorem
All systems (DGF) that satisfy condition (GPE) are GAS if andonly if (1.8) holds.
We stress that our interest lies in the study of systems satisfying (GPE) as a class.That is, the above theorem states that if (1.8) is not satisfied, then there exists aninput signal satisfying (GPE) that is not GAS. However, for a fixed signal satisfying(GPE), condition (1.8) is not necessary for GAS, as shown in [6, Prop. 7].
We now turn to a brief description of the strategy ofproof. The main idea is to consider optimal control problems whose minimal valuesprovide bounds for the worst-rate of exponential decay.More precisely, since the dynamics in (DGF) are linear in x ∈ R n , the systemis amenable to be decomposed in spherical coordinates. Thus, letting x = rω , for r = k x k ∈ R + and ω = x/ k x k ∈ S n − , (DGF) reads as˙ r = − rω ⊤ Sω, (1.9) ˙ ω = − Sω + ( ω ⊤ Sω ) ω. (1.10)For S satisfying (PE), consider the control system defined by (1.5) and let Φ S ( · , · )be the fundamental matrix associated with S , i.e., for every 0 ≤ s ≤ t , Φ S ( t, s ) isthe value at time t of the solution of ˙ M = − SM with initial condition M ( s ) = Id n .Observe that, for every x ∈ R n \ { } , if we let Φ S ( t, x = r ( t ) ω ( t ), then it holds(1.11) ln (cid:18) k Φ S ( T + t, x kk Φ S ( t, x k (cid:19) = ln (cid:18) r ( t + T ) r ( t ) (cid:19) = − Z t + Tt ω ⊤ Sω ds, ∀ t ≥ . Since the last term in the above equation does not depend on r , this suggests toconsider the optimal control problem(OCP) inf J ( S, ω ) , J ( S, ω ) := Z T ω ⊤ Sω dt, where the infimum is considered among all signals satisfying(INT) a Id n ≤ Z T S ( τ ) dτ ≤ b Id n , and initial conditions ω ∈ S n − , and ω : [0 , T ] → S n − is the trajectory of (1.10)with initial condition ω (0) = ω . In particular, S satisfies (INT) if and only if it isthe restriction to [0 , T ] of a signal satisfying (PE).We show in Proposition 2.2 below that (OCP) admits minimizers and that thecorresponding minimal value is independent of T . Denoting by µ ( a, b, n ) this value,in Section 3 we reduce the proof of the main results to the following. roposition There exists a universal constant C > such that, for every < a ≤ b T > and integer n ≥ , (1.12) µ ( a, b, n ) ≤ C a b . Moreover, there exists a T -periodic rank-one control S ∗ = c ∗ c ⊤∗ such that a Id n ≤ Z t +2 Tt S ∗ ( τ ) dτ ≤ b Id n , ∀ t ≥ , and an initial condition ω ∈ S n − such that (1.13) ω ∗ ( t ) = Φ S ∗ ( t, ω k Φ S ∗ ( t, ω k is a T -periodic trajectory and both t S ∗ | [0 ,T ] ( t ) and t S ∗ | [ T, T ] ( t − T ) , togetherwith the respective initial conditions ω and ω ∗ ( T ) , are minimizers for (OCP) .Remark J ( S, ω ) restrictedto controls of the form S = cc ⊤ is still equal to µ ( a, b, n ). The above propositionprovides the stronger conclusion that µ ( a, b, n ) is actually attained by a rank-oneminimizer.The rest of the paper is devoted to prove the above proposition. We first observethat, due to the monotonicity with respect to the dimension n of the minimal value µ ( a, b, n ), for the first part of the statement it is enough to bound µ ( a, b, n ≥ We use ⌊ x ⌋ to denote the integer part of the real number x and J a, b K to denote the set of integers in [ a, b ]. We let Sym n be the set of n × n symmetricreal matrices, and by Sym + n the subset of non negative ones. Moreover, for a ≤ b , weuse Sym n ( a, b ) to denote the set of matrices Q ∈ Sym n such that a Id n ≤ Q ≤ b Id n in the sense of quadratic forms. For every positive integer k , we denote by S k theunit sphere of R k +1 . Finally, we let Sym + n ( a, b, T ) be the set of functions that satisfy(INT). Similarly, we let Sym ( P E ) n ( a, b, T ) be the set of functions that satisfy (PE).
2. Preliminary results for the optimal control problem (OCP) . We startby showing a simple upper bound for the minimal value µ ( a, b, T, n ) of (OCP). Proposition
It holds (2.1) µ ( a, b, T, n ) ≤ a. Proof.
It suffices to consider the matrix S defined by(2.2) S ( t ) = anT e j e ⊤ j if t ∈ (cid:20) ( j − Tn , jTn (cid:19) , j = 1 , . . . , n, where { e , . . . , e n } ⊂ R n denotes the canonical basis of R n . Indeed,(2.3) Z T S ( t ) dt = a Id n so that S satisfies (INT), and, for ω = e , we have that ω ≡ ω and J ( S, ω ) = a . ince we want to apply techniques of optimal control to study the minimal value µ ( a, b, T, n ) of (OCP), we now establish existence of minimizers for such problem. Proposition
The optimal control problem (OCP) admits minimizers withconstant trace. Moreover, the minimal value µ ( a, b, T, n ) is independent of T > .Proof. In order to prove the first part of the proposition, we first notice that theinfimum in (OCP) remains unchanged if we assume that S ( t ) > t ∈ [0 , T ].Indeed any S satisfying (INT) may be approximated arbitrarily well by a positivedefinite signal S ε = aa + ε ( S + ε Id n /T ) with ε > J depends continuously on the control S (e.g., in the L topology) andon the initial condition ω (this may be easily deduced from the continuous dependenceon S and x (0) of the original equation (DGF)).We now show that for any S > S ofconstant trace satisfying (INT) and such that J ( ˜ S, ω ) = J ( S, ω ). Setting T = R T Tr( S ( t )) dt, we consider the change of time τ ( t ) = T T ∫ t Tr( S ( s )) ds , which is welldefined from [0 , T ] to itself since Tr( S ( t )) > t ∈ [0 , T ]. If x is the solutionof (DGF) with control S it is then easy to see that ˜ x = x ◦ τ − solves (DGF) withcontrol(2.4) ˜ S ( · ) = T S ( τ − ( · )) T Tr( S ( τ − ( · ))) , so that J ( ˜ S, ω ) = J ( S, ω ), and moreover R T ˜ S ( s ) ds = R T S ( s ) ds .Note now that the set of matrix-valued functions of constant trace in Sym + n ( a, b, T )weakly- ∗ compact in L ∞ (see Lemma A.1 in Appendix A). The existence of minimizerswith constant trace is then a consequence of the continuous dependence of the func-tional J on S and ω , which in turn may be deduced from the continuous dependenceon S (in the weak- ∗ topology of L ∞ ) and x (0) of the solutions of (DGF).Finally, the independence of µ ( a, b, T, n ) from T may be deduced from the factthat, given a solution x of (DGF) corresponding to S ∈ Sym + n ( a, b, T ), any timereparametrization ˜ x of x defined on [0 , ˜ T ] is the solution of (DGF) for some ˜ S ∈ Sym + n ( a, b, ˜ T ). Remark µ ( a, b, n ) = µ ( a, b, T, n ).The following observation will be crucial in the sequel. Proposition
The map n µ ( a, b, n ) is non-increasing.Proof. Consider an admissible trajectory ω of (DGF) in dimension n , associatedwith some S ∈ Sym + n ( a, b, T ) and ω ∈ S n − . Then, the trajectory ˜ ω = ( ω,
0) is atrajectory of (DGF) in dimension m ≥ n associated with ˜ S = diag( S, a Id m − n ) ∈ Sym + m ( a, b, T ) and initial condition ˜ ω = ( ω , ω , we triviallyhave that J ( S, ω ) = J ( ˜ S, ˜ ω ), and thus µ ( a, b, n ) ≥ µ ( a, b, m ).
3. Reduction of the main results to Proposition 1.6.
In this section, weshow that Theorems 1.1, 1.3, and 1.5, all follow from Proposition 1.6. To this aim,we start by determining the homogeneity with respect to T of the quantities at hand. Proposition
For every
T > it holds (3.1) R ( a, b, T, n ) = R ( a, b, , n ) T and γ ( a, b, T, n ) = T γ ( a, b, , n ) roof. If S ∈ Sym ( P E ) n ( a, b, T ), then letting ˜ S ( s ) := T S ( T s ), we have thatΦ ˜ S ( s,
0) = Φ S ( T s,
0) for all s > S ∈ Sym ( P E ) n ( a, b, x ( · ) is the trajectory of(1.5) associated with S and u ∈ L ((0 , + ∞ ) , R n ), then x ( T · ) is associated with ˜ S and˜ u ( s ) := T u ( T s ). This yields at once that γ ( c ) = T γ (˜ c ), completing the proof.We are now ready to establish the link between the minimal value µ ( a, b, n ) of(OCP) and the worst rate of exponential decay for (DGF). We observe that thisyields at once the fact that Theorem 1.1 is a consequence of Proposition 1.6. Proposition
It holds that, (3.2) µ ( a, b, n ) T ≤ R ( a, b, T, n ) ≤ µ ( a/ , b/ , n ) T Moreover, the same result holds true when replacing R ( a, b, T, n ) by the quantity ob-tained by restricting (1.1) to rank-one matrices (i.e., S = cc ⊤ and c ∈ R n ).Proof. Thanks to Proposition 3.1, we can restrict to the case T = 1. Let ( S l ) l ≥ ⊂ Sym ( P E ) n ( a, b,
1) be a minimizing sequence for R ( a, b, , n ), i.e., such that there existsa vanishing sequence of positive numbers ( ε l ) l ≥ satisfying R ( S l ) ≤ R ( a, b, , n ) + ε l for l ≥
0. By the definition of the object at hand, there exists an increasing sequence( t l ) l ≥ of times tending to infinity and a sequence ( ω l ) l ≥ of unit vectors such that,for every l ≥
0, it holds(3.3) ln k Φ S l ( t l , ω l k = ln k Φ S l ( t l , k ≥ ( − R ( S l ) − ε l ) t l . Fix l ≥ k := ⌊ t l ⌋ , y := Φ S l ( t l − k, ω l , y j +1 := Φ S l ( t l − ( k − j − , t l − ( k − j )) y j , ≤ j ≤ k − . From (1.11), we then get(3.5) ln (cid:18) k Φ S l ( t l , ω l kk y k (cid:19) = k − X j =0 ln (cid:18) k y j +1 kk y j k (cid:19) ≤ − kµ ( a, b, n ) . Clearly, there exists a positive constant K ≤ l ≥ K ≤k y k ≤
1. Thus, since ( t l ) l ≥ is unbounded, we deduce at once that(3.6) − R ( a, b, , n ) − ε l ≤ − kµ ( a, b, n ) /t l . By letting l tend to infinity, this yield the l.h.s. of (3.2).The r.h.s. of (3.2) will follow from the inequality R (2 a, b, , n ) ≤ µ ( a, b, n ) to beproved next. Let S ∗ = c ∗ c ⊤∗ ∈ Sym ( P E ) n (2 a, b,
2) be the 2-periodic control given byProposition 1.6 for T = 1. It then follows from the latter and (1.11) that(3.7) ln k Φ S ∗ ( k, k = k X ℓ =1 ln k Φ S ∗ ( ℓ, ℓ − k = − kµ ( a, b, n ) , k ∈ N . Then, standard arguments yield(3.8) R (2 a, b, , n ) ≤ R ( S ∗ ) ≤ − lim ℓ → + ∞ ln k Φ S ∗ (2 ℓ, k ℓ = µ ( a, b, n ) , concluding the proof. he following links the L -gain γ ( a, b, n, T ) with the minimal value of (OCP). Proposition
For every < a ≤ b , T > and integer n ≥ , one has (3.9) T µ ( a/ , b/ , n ) ≤ γ ( a, b, n, T ) ≤ T − e − µ ( a,b,n ) . Proof.
Thanks to Proposition 3.1 it suffices to consider the case T = 1. We startby establishing the right-hand side inequality of (3.9). From the variation of constantformula, for every control u ∈ L ((0 , + ∞ ) , R n ), cc ⊤ ∈ Sym ( P E ) n ( a, b,
1) and t ≥
0, thesolution of (1.5) with x u (0) = 0 reads(3.10) x u ( t ) = Z t Φ cc ⊤ ( t, s ) u ( s ) ds, t ≥ . Since it is easy to deduce from the definition of µ := µ ( a, b, n ) that k Φ cc ⊤ ( t, s ) k ≤ e − µ ⌊ t − s ⌋ for every t ≥ s ≥
0, the above implies(3.11) k x u ( t ) k ≤ Z t e − µ ⌊ t − s ⌋ k u ( s ) k ds, t ≥ . Let h be the characteristic function of R + . Define on R the function f ( s ) = e − µ ⌊ s ⌋ h ( s ),which is square integrable over R + . Then the r.h.s. of (3.11) is equal to the convolutionproduct of f and k u ( · ) k . By convolution and Plancherel theorems, one has that the L -gain of (1.5) is upper bounded by k F k ∞ where F is the Fourier transform of f .It is now straightforward to observe that the supremum of F is attained at 0, whichyields the desired upper bound.We next give an argument for the the left-hand side inequality of (3.9). For thatpurpose, consider S ∗ = c ∗ c ⊤∗ ∈ Sym ( P E ) n (2 a, b,
2) and ω ∗ ∈ S n − as provided byProposition 1.6 for T = 1. For t ∈ [0 , ρ ( t ) := k Φ S ∗ ( t, ω ∗ k , ˆ ρ := ρ (2) = exp( − µ ) < . Since S ∗ is 2-periodic, one has that for every t ≥ s ≥ k ≥ l ,(3.13) Φ S ∗ ( t + 2 l, s + 2 l ) = Φ S ∗ ( t, s ) , Φ S ∗ (2 k, l ) ω ∗ = ˆ ρ k − l ω ∗ . For t ≥
0, set k t := ⌊ t/ ⌋ and ξ t = t − k t , i.e., t = 2 k t + ξ t with ξ t ∈ [0 , v defined on [0 , u : R + → R n given by(3.14) u ( t ) = v ( ξ t )Φ S ∗ ( ξ t , ω ∗ , t ≥ . Observe that u is 2-periodic. Let x u be the trajectory of (1.5) associated with u andstarting at the origin. Then, by using (3.10) and (3.13), one has, for t ≥ x u ( t ) = Z t k t Φ S ∗ ( t, s ) u ( s ) ds + Φ S ∗ ( t, k t ) k t − X j =0 Z j +1)2 j Φ S ∗ (2 k t , s ) u ( s ) ds = Z ξ t Φ S ∗ ( ξ t , s ) u ( s ) ds + Φ S ∗ ( ξ t , k t − X j =0 Z Φ S ∗ (2( k t − j ) , s ) u ( s ) ds. (3.15) et V ( t ) := R t v ( s ) ds for t ∈ [0 , x u ( t ) = (cid:16) V ( ξ t ) + ˆ ρ − ˆ ρ (1 − ˆ ρ k t ) V (2) (cid:17) Φ S ∗ ( ξ t , ω ∗ , t ≥ . Thus, for every positive integer k we have(3.17) Z k k x u ( t ) k dt = k Z z ( t ) ρ ( t ) dt + r k , where z is the function defined by(3.18) z ( t ) = V ( t ) + ˆ ρ − ˆ ρ V (2) , t ∈ [0 , , and | r k | ≤ C P k − j =0 ˆ ρ j ≤ C/ (1 − ˆ ρ ) for some positive constant C . On the other hand,(3.19) Z k k u ( t ) k dt = k Z v ( t ) ρ ( t ) dt. For every positive integer k , let u k be the input function defined as follows: it isequal to u on [0 , k ] and zero elsewhere. We use x k to denote the trajectory of (1.5)associated with u k and starting at the origin. Note that x k ( t ) = Φ c ∗ ( t − k, x u (2 k ),for t ≥ k , which decreases exponentially to zero as t tends to infinity. Then, we have(3.20) γ (2 a, b, n, ≥ lim sup k →∞ k x k k L k u k k L = vuut R z ( t ) ρ ( t ) dt R v ( t ) ρ ( t ) dt , ∀ v . By Proposition 3.1 we have γ (2 a, b, n,
2) = 2 γ (2 a, b, n, µ given in Proposition 1.6, in order to complete the proof of Theorem 1.3,it suffices to show that there exists v z ( t ) = v ( t ) /µ = V ′ ( t ) /µ for all t ≥
0. By definition of z , such a function v exists if and only if there exists C = 0such that the nonzero solution of the equation(3.21) 1 µ V ′ ( t ) = V ( t ) + C, V (0) = 0satisfies ˆ ρ − ˆ ρ V (2) = C . By taking into account (3.12), it is easy to show that this isthe case.As a consequence of the previous result, of Proposition 1.6, and of Theorem 1.1,we now prove Theorem 1.3. Proof of Theorem 1.3.
The left-hand side of (1.7) is a consequence of the left-hand side of (3.9) together with (1.12). Regarding the proof of the right-hand sideof (1.7), we first notice that (1.12) implies that µ ( a, b, n ) ≤ C /
2. As a consequenceof the monotonicity of x x − e − x we then get11 − e − µ ( a,b,n ) = µ ( a, b, n )1 − e − µ ( a,b,n ) µ ( a, b, n ) ≤ C µ ( a, b, n ) , where C = C − e − C / ) . By using the right-hand side of (3.2) and (1.3) we thus obtain(3.22) γ ( a, b, n, T ) ≤ T C µ ( a, b, n ) ≤ C R (2 a, b, T, n ) ≤ C C (1 + nb ) Ta , concluding the proof. e finally prove Theorem 1.5, relying on the validity of Proposition 1.6. Proof of Theorem 1.5.
It is enough to prove that the condition provided in thestatement of the theorem is a necessary condition for (GAS). Consider the threesequences ( a l ) l ≥ , ( b l ) l ≥ and ( τ l ) l ≥ verifying the assumptions of the theorem. Forevery l ≥
1, we define T l := τ l +1 − τ l and apply Proposition 1.6 to ( a l , b l , T l ) to deducethat there exists S l in Sym + n ( a l , b l , T l ) and ω l ∈ S n − such that J ( S l , ω l ) = µ ( a l , b l , T l )and the trajectory of (1.10) starting at ω l and corresponding to S l is 2 T -periodic.Choose a sequence ( U l ) l ≥ in O( n ) such that, if Σ l is the function defined on [0 , τ l ]as the concatenation of the U j S j , 0 ≤ j ≤ l − y j ) ≤ j ≤ l − is the sequencedefined by y := ω and y j +1 := Φ Σ l ( τ j +1 , τ j ) y j , then one has, for 0 ≤ j ≤ l −
1, that(3.23) k y j +1 kk y j k = k Φ S j ( T j , w k . By summing up these relations and using the definitions of the objects at hand, oneobtains(3.24) − ln k Φ Σ l ( τ l , ω k = l − X j =0 µ ( a j , b j , n ) . From Proposition 1.6, one deduces that the series of general term µ ( a l , b l , n ) convergesif and only if the series of general term a l b l converges. Together with the aboveequation, one easily concludes.
4. Existence of rank one periodic minimizers for (OCP) . In this section,we prove the second part of Proposition 1.6. This is done via the following.
Proposition
There exists a rank-one S ∗ = c ∗ c ⊤∗ ∈ Sym ( P E ) n ( a, b, T ) and aninitial condition ω ∈ S n − such that (4.1) ω ∗ ( t ) = Φ c ∗ ( t, ω k Φ c ∗ ( t, ω k , is T -periodic and both t S ∗ | [0 ,T ] ( t ) and t S ∗ | [ T, T ] ( T − t ) , together with theirrespective initial conditions ω and ω ∗ ( T ) , are minimisers for (OCP) . In order to prove the above, we apply the Pontryagin Maximum Principle (PMPfor short) to the minimizer with constant trace of (OCP) given by Proposition 2.2.As usual, in order to get rid of the constraint (INT) we introduce an auxiliaryvariable Q ∈ Sym n , and reformulate (OCP) as follows: Minimize J ( S, ω ) with respectto S ∈ Sym + n ( a, b, T ) and ω ∈ S n − along trajectories of˙ ω = − Sω + (cid:0) ω T Sω (cid:1) ω, (4.2) ˙ Q = S, (4.3)starting at ( ω , Q ( T ) ∈ Sym n ( a, b ). The state space of the system is M = S n − × Sym n . We will henceforth identify the cotangent space at ( ω, Q ) ∈ M with T ∗ ω S n − × T ∗ Q Sym n ≃ ( R ω ) ⊥ × Sym n .According to the PMP, a solution ( ω, Q ) of the optimal control problem (OCP)is necessarily the projection of an extremal , i.e., an integral curve λ ∈ T ∗ M of theHamiltonian vector on T ∗ M satisfying certain additional conditions. We hereby pres-ent a definition of extremal adapted to our setting. The fact that this is equivalentto the standard definition of extremal is the subject of the subsequent proposition. efinition A curve λ : [0 , T ] → T ∗ M is an extremal with respect to thecontrol S ∈ Sym + n ( a, b, T ) and ω ∈ S n − if:(i) letting λ = ( ω, Q, p, P Q ) , it satisfies ˙ ω = − Sω + (cid:0) ω T Sω (cid:1) ω, (4.4) ˙ Q = S, (4.5) ˙ p = Sp − ( ω ⊤ Sω ) p − ˙ ω, (4.6) ˙ P Q = 0 . (4.7) (ii) It holds that p (0) = p ( T ) = 0 and that − P Q belongs to the normal cone of Sym n ( a, b ) at Q ( T ) .(iii) Let (4.8) M := P Q − ( ωp ⊤ + pω ⊤ + ωω ⊤ ) on [0 , T ] . Then, M ≤ and M S = SM ≡ on [0 , T ] . Note that, by conditions ( i ) , ( ii ) in Definition 4.2, p T ω ≡ T ∗ ω S n − ≃ ( R ω ) ⊥ . We then get the following. Proposition
Let ( ω, Q ) : [0 , T ] → M be an optimal trajectory of the optimalcontrol problem (OCP) , whose optimal control S has constant trace. Then ( ω, Q ) isthe projection on M of an extremal λ : [0 , T ] → T ∗ M .Proof. Recall that the existence of ( ω, Q ) : [0 , T ] → M as an optimal trajectoryassociated with a control S of constant trace is guaranteed by Proposition 2.2. Aftersome computations, deferred to Proposition B.2 in Appendix B, the PMP implies thatthere exists a curve t ∈ [0 , T ] ( p ( t ) , P Q ( t )) and ν ∈ { , } with ( p ( t ) , P Q ( t ) , ν ) = 0a.e. on [0 , T ] such that1. ( p ( t ) , P Q ( t )) ∈ T ∗ ω ( t ) S n − × T ∗ Q ( t ) Sym n satisfy on [0 , T ] the adjoint equations:˙ p = Sp − ( ω ⊤ Sω ) p − ν ˙ ω, (4.9) ˙ P Q = 0;(4.10)2. letting λ ( t ) = ( ω ( t ) , Q ( t ) , p ( t ) , P Q ) we have the maximality condition:(4.11) H ( λ ( t ) , S ( t )) = max S ∈ Sym + n H ( λ ( t ) , S ) a.e. on [0 , T ] , where H ( ω, Q, p, P Q , S ) = Tr( S ˜ M ) / M ∈ Sym n is defined by(4.12) ˜ M = P Q − (cid:0) ωp ⊤ + pω ⊤ + ν ωω ⊤ (cid:1) , with ν ∈ { , } .
3. we have the transversality conditions:(4.13) p (0) ⊥ T ∗ ω (0) S n − , p ( T ) ⊥ T ∗ ω ( T ) S n − , and − P Q belongs to the normal cone of Sym n ( a, b ) at Q ( T ).Note that Item (ii) of Definition 4.2 is equivalent to the transversality conditions,since p ∈ T ∗ ω S n − by definition. We are left to prove Items (i) and (iii) . For thispurpose, we claim that the maximality condition implies that(4.14) H ( λ ( t ) , S ( t )) ≡ M ≤ , T ] . ndeed, H ( λ ( t ) ,
0) = 0 and if there exists ¯ S ∈ Sym + n such that H ( λ ( t ) , ¯ S ) > H ( λ ( t ) , γ ¯ S ) → + ∞ as γ → + ∞ , provingthe first part of the claim. As a consequence, Tr( S ˜ M ) ≤ S ∈ Sym + n .In particular, for any z ∈ R n , Tr( zz ⊤ ˜ M ) = z ⊤ ˜ M z ≤
0, which shows that ˜ M ≤ S ˜ M ) ≤ S ˜ M = ˜ M S = 0. Let us now prove that ν = 1, which will yield at once Items (i) and (iii) . We argue by contradiction and assume ν = 0. In this case, (4.9) is alinear ODE and, due to Item (ii) , its solution is p ≡
0. This and (4.14) imply that P Q ≤ P Q S ) ≡
0. Hence, P Q S ≡
0. Integrating over [0 , T ] this relation yields P Q Q ( T ) = 0, which implies P Q = 0 since Q ( T ) ≥ a Id and hence is invertible. This,however, contradicts the fact that ( p, P Q , ν ) = 0, thus showing that ν = 1.We will also need the following. Proposition
Let λ = ( ω, Q, p, P Q ) be an extremal with respect to an optimalcontrol S . Then, up to an orthonormal change of basis, there exists k, r ∈ N , with n = 1 + k + r , α ∈ (0 , , and positive definite diagonal matrices D Q ∈ R r × r and D b ∈ R k × k , with all elements of D Q belonging to the interval [ a, b ] , such that (4.15) Q ( T ) = diag( a, b Id k , D Q ) and P Q = diag( α, − D b , r ) . Proof.
Since P Q ≤ ω ω ⊤ , one deduces at once that λ max ( P Q ) ≤ λ max ( ω ω ⊤ ) = 1.We now claim that P Q has exactly one positive eigenvalue α ∈ (0 , P Q has at most one positive eigenvalue. Indeed, by Item (iii) of Definition 4.2, P Q − ω ω ⊤ is negative semi-definite. Therefore, the restrictionof the quadratic form defined by P Q to ( R ω ) ⊥ is also negative semi-definite. Thisimplies that P Q has at least n − P Q cannot be negative semi-definite. Arguing by contradiction,one has that for every t ≤ t in [0 , T ],(4.16) Tr (cid:16) P Q (cid:0) Q ( t ) − Q ( t ) (cid:1)(cid:17) ≤ . Let T ≤ T be the largest time in [0 , T ] such that Sω ≡ p ≡ , T ]. Wefirst prove that T exists and is strictly positive. For that purpose, pick ¯ t ∈ [0 , T ] suchthat k p (¯ t ) k = max {k p ( s ) k | ≤ s ≤ ¯ t } and Tr( Q (¯ t )) ≤ /
2. One deduces that(4.17) Z ¯ t p ⊤ Sp dt ≤ max s ∈ [0 , ¯ t ] k p ( s ) k Z ¯ t λ max ( S ) dt ≤ k p (¯ t ) k Tr( Q (¯ t )) ≤ k p (¯ t ) k . We next prove the following two equalities, holding for every t ≤ t in [0 , T ],12 (cid:0) k p ( t ) k − k p ( t ) k (cid:1) + Z t t ( ω T ( t ) S ( t ) ω ( t )) k p ( t ) k dt = Z t t p T ( t ) S ( t ) p ( t ) dt + Z t t p T ( t ) S ( t ) ω T ( t ) dt, and(4.18) Tr (cid:16) P Q (cid:0) Q ( t ) − Q ( t ) (cid:1)(cid:17) = 2 Z t t p T ( t ) S ( t ) ω T ( t ) dt + Z t t ω T ( t ) S ( t ) ω ( t ) dt. Both equalities follow by Proposition 4.3: for the first one, we multiply by p ⊤ thedynamics of ˙ p given by (4.6) and integrate it on [ t , t ] using the fact that p ⊤ ω = 0.We integrate Tr( M S ) over [ t , t ], with M given in (4.8), to obtain the second one. t is immediate to deduce from (10) and (4.18) that, for every t ≤ t in [0 , T ],(4.19) Z t t p T ( t ) S ( t ) p ( t ) dt = 12 Z t t ω T ( t ) S ( t ) ω ( t ) dt + 12 (cid:0) k p ( t ) k − k p ( t ) k (cid:1) + Z t t ( ω T ( t ) S ( t ) ω ( t )) k p ( t ) k dt −
12 Tr (cid:16) P Q (cid:0) Q ( t ) − Q ( t ) (cid:1)(cid:17) . By using (4.16),(4.17) and (4.19) with t = 0 and t = ¯ t , one deduces that(4.20) Z ¯ t ω ⊤ Sω dt ≤ Tr( P Q Q (¯ t )) ≤ . This immediately implies that Sω ≡ p ≡ , ¯ t ], proving the existence of T asclaimed. Note that, necessarily T < T , since otherwise one would have that ω ≡ ω on [0 , T ] and, integrating Sω ≡ , T ] would yield that Q ( T ) ω = 0, contradictingthe fact that Q ( T ) ≥ a Id.We next pick T + ¯ t ∈ [0 , T ] such that k p ( T + ¯ t ) k = max {k p ( s ) k | T ≤ s ≤ T + ¯ t } and Tr( Q ( T + ¯ t )) − Tr( Q ( T )) ≤ /
2. We then reproduce the argument starting in(4.17) where we replace the pair of times (0 , ¯ t ) by the pair of times ( T , T + ¯ t ). Inthat way, we extend the interval on which both Sω and p are zero beyond T , hencecontradicting the definition of T . We have completed the argument for the existenceof a unique positive eigenvalue α for P Q .We are left to show that P Q and Q ( T ) can be put in the form (4.15) by anorthonormal change of basis. By definition − P Q belongs to the normal cone ofSym n ( a, b ) at Q ( T ) if and only if Tr( P Q ( ˆ Q − Q ( T ))) ≥ Q ∈ Sym n ( a, b ).Assume without loss of generality that Q ( T ) is diagonal and let λ i , i = 1 , . . . , n , bethe eigenvalue of Q ( T ) corresponding to the eigenvector e i of the canonical basis. If λ i ∈ ( a, b ), then it is easy to check that the matrices ˆ Q ± = Q ( T ) ± ε ( e i e Tj + e j e Ti ),for j = 1 , . . . , n , belong to Sym n ( a, b ) if ε > P Q ( ˆ Q + − Q ( T ))) ≥ P Q ( ˆ Q − − Q ( T ))) ≥ i, j ) component of P Q must be 0. If λ i = a then Tr( P Q ( ˆ Q − Q ( T ))) ≥
0, withˆ Q = Q ( T ) + εe i e Ti ∈ Sym n ( a, b ), implies that the component ( i, i ) of P Q is nonpos-itive. Similarly one deduces that λ i = b implies that the component ( i, i ) of P Q isnonnegative. Consider now any two eigenvalues a ≤ λ i < λ j ≤ b of Q ( T ). Then it iseasy to check that the matrices ˆ Q ± = Q ( T ) ± ε ( e i e Tj + e j e Ti ) + kε ( e i e Ti − e j e Tj )with k > / ( λ j − λ i ) belong to Sym n ( a, b ) if ε is small enough. Again, sinceTr( P Q ( ˆ Q ± − Q ( T ))) ≥
0, and letting ε tend to zero, one gets that the ( i, j ) com-ponent of P Q must be zero. One deduces that P Q commutes with Q ( T ) and the twomatrices can thus be simultaneously diagonalized, taking the form (4.15). Proposition
Let λ = ( ω, Q, p, P Q ) be an extremal of (OCP) associated witha control S . Assume that, in the notations of Proposition 4.4, one has that r ≥ .Then, there exist (˜ ω, ˜ p ) ∈ T ∗ S k , ˜ S ∈ Sym + k +1 ( a, b, T ) and S ∈ Sym + r ( a, b, T ) , suchthat (4.21) ω = (˜ ω, ⊤ , p = (˜ p, ⊤ and S = diag( ˜ S, S ) . Moreover, letting ˜ Q = diag( a, b Id k ) and ˜ P Q = diag( α, − D b ) , we have that ˜ λ =(˜ ω, ˜ Q, ˜ p, ˜ P Q ) is an extremal trajectory with control ˜ S of (OCP) in dimension k + 1 ,and J ( ˜ S, ˜ ω (0)) = J ( S, ω (0)) . In particular, if k = 0 , then p ≡ and there exists ω ∈ S n − such that ω ≡ ω and J ( S, ω ) = a . roof. We start by decomposing ω = (˜ ω, ξ ) and p = (˜ p, q ) for some R r -valuedfunctions ξ and q . Our aim is to prove that q ≡ ξ ≡
0. Let us define˜ A = (˜ p + ˜ ω )(˜ p + ˜ ω ) ⊤ − ˜ p ˜ p ⊤ , A = ( q + ξ )( q + ξ ) ⊤ − qq ⊤ (4.22) B = (˜ p + ˜ ω )( q + ξ ) ⊤ − ˜ pq ⊤ (4.23)Then, by Item ( iii. ) of Definition 4.2, we get(4.24) (cid:18) ˜ A − ˜ P Q BB ⊤ A (cid:19) ≥ . We deduce at once that A ≥
0, and thus, that there exists ̺ ∈ [ − ,
1] such that q = ̺ ( q + ξ ). In particular, q + ξ = 0 if and only if q = ξ = 0. Let I be a maximalopen interval such that q + ξ = 0 and assume, by contradiction, that I = ∅ .We claim that(4.25) − (1 − ̺ ) ˜ P Q ≥ (cid:0) (1 − ̺ )˜ p + ̺ ˜ ω (cid:1)(cid:0) (1 − ̺ )˜ p + ̺ ˜ ω (cid:1) ⊤ on I. To this effect, set A ε = A + ε ( q + ξ )( q + ξ ) ⊤ for ε >
0. Observe that (4.24) holdswith A replaced by A ε . Then, by Schur complement formula we have(4.26) ˜ A − ˜ P Q − BA † ε B ⊤ ≥ , where we denoted by A † ε the Moore-Penrose inverse of A ε . Let us observe that(4.27) A ε = (1 − ̺ + ε )( q + ξ )( q + ξ ) ⊤ and B = ((1 − ̺ )˜ p + ˜ ω ) ( q + ξ ) ⊤ . Since A † = ( q + ξ )( q + ξ ) ⊤ (1 − ̺ + ε ) k q + ξ k , the claim follows by letting ε ↓ ̺ = 1.Indeed, by (4.25), we have − (1 − ̺ ) α ≥ α >
0. On the other hand, if ̺ = 1,we have ξ ≡ I by definition of ̺ , and ˜ ω ≡ I by (4.25), which contradicts ω ∈ S n − . Thus, ̺ = − p + ω ≡ I . However,since p ∈ ω ⊥ , we have k p + ω k ≥ k ω k ≡
1, thus yielding the desired contradiction.This implies that I = ∅ , and thus that ξ ≡ q ≡ , T ].Setting S = (cid:18) ˜ S S D S ⊤ D S (cid:19) , it is easy to check from (4.4) and (4.6) that ˜ ω ⊤ S D ≡ p ⊤ S D ≡
0. Then, it follows from Item ( iii ) of Definition 4.2 that ˜ P Q S D ≡
0, sothat we can conclude that S D ≡
0. This yields the desired form for S , together withthe fact that ˜ λ is an extremal trajectory with control ˜ S of (OCP) in dimension k + 1satisfying J ( ˜ S, ˜ ω (0)) = J ( S, ω (0)). Finally, the last part of the statement follows bythe explicit computation of the solutions in dimension n = 1.As we will see, the above Proposition immediately yields Proposition 4.1 if we arein the case k = 0. Thus, we henceforth focus on extremals that satisfy the following. Assumption λ = ( ω, Q, p, P Q ) is such that(4.28) Q ( T ) = diag( a, b Id n − ) and P Q = diag( α, − D b ) , where α ∈ (0 , D b ∈ R ( n − × ( n − is a positive definite diagonal matrix. e start by proving some essential properties of the matrix(4.29) M = P Q − (cid:0) ωp ⊤ + pω ⊤ + ωω ⊤ (cid:1) . We recall that, by Item ( iii. ) of Definition 4.2, we have(4.30) M ≤ M S ≡ SM ≡ . Proposition
Let λ be an extremal satisfying Assumption 4.6. Then, rank M ≡ n − , and M has constant spectrum (taking into account multiplicities). In particular, rank S ≡ .Proof. By (4.8), we have rank M ≤ n −
1. Let x ∈ ker M , x = 0, then(4.31) x = (cid:0) ( p + ω ) ⊤ x (cid:1) P − Q ω + ( ω ⊤ x ) P − Q p. As a consequence, it holds ker M ⊂ V := span { P − Q ω, P − Q p } . Observe that dim V ∈{ , } , with dim V = 2 if and only if p = 0. In particular, rank M ≥ n − E = { t ∈ [0 , T ] | rank M ( t ) = n − } . Trivially, E is open in [0 , T ], and,moreover, E = ∅ , since { , T } ⊂ E . We now show that E is also closed, whichimplies E = [0 , T ] thus completing the proof of the statement.To this aim, let us start by observing that, thanks to (4.30), on E it holds S = cc ⊤ for some c ∈ ker M . By Proposition 2.2, one can assume that c has constant norm.We now claim that t ∈ E c ( t ) is an analytic function of ( ω ( t ) , p ( t )). To see that,set α = ( p + ω ) ⊤ x and β = ω ⊤ x . By multiplying (4.31) by ( p + ω ) ⊤ and ω ⊤ , oneobserves that for all t ∈ E the vector ( α ( t ) , β ( t )) belongs to the kernel of a 2 × ω ( t ) , p ( t )). Theclaim follows by using the fact that c has constant norm. By the dynamics (4.4)-(4.6), this implies that the extremal trajectory and t M ( t ) are analytic on E . As aconsequence, the (unordered) negative eigenvalues ( − λ j ) n − j =1 of M are analytic on E ,and it is possible to find an analytic family { v , . . . , v n − } of associated orthonormaleigenvectors, (see, e.g., [12, Theorem 6.1 and Section 6.2]).Differentiating with respect to t ∈ E the relation v ⊤ ℓ M v ℓ = λ ℓ for ℓ = 1 , . . . , n − v ⊤ ℓ ˙ M v ℓ = ˙ λ ℓ on E. To complete the proof of the statement, observe that straightforward computationsfrom the definition of M yield(4.33) ˙ M = ( ω ⊤ c )( cp ⊤ + pc ⊤ ) − ( p ⊤ c )( cω ⊤ + ωc ⊤ ) on E. Using this and the fact that c ⊤ v ℓ ≡ λ ℓ ≡ ℓ = 1 , . . . , n − E is closed, completing the proof of the statement. Proposition
Let λ be an extremal satisfying Assumption 4.6. Then, (4.34) ω i (0) = ω i ( T ) for i = 1 , . . . , n Proof.
Let us write P Q = diag( − d , . . . , − d n ), where d = α and 0 < d i ≤ d i +1 ,for 2 ≤ i ≤ n . By (4.30) and Proposition 4.7, the control associated with λ takes theform S = cc ⊤ for some vector valued function c
0. Set(4.35) q = p + ω/ , γ = c ⊤ ω, δ = c ⊤ q. hen, the fact that M c ≡ c i = − d i − (cid:0) ( c ⊤ q ) ω i + ( c ⊤ ω ) q i (cid:1) , i = 1 , . . . , n. Moreover, since by Assumption 4.6 we have Q ( T ) = diag( a, b Id n − ), this implies(4.37) Z T c c i dt = aδ i and Z T c i c j dt = bδ ij , i, j = 2 , . . . , n. Finally, letting Z i = ( ω i , q i ) ⊤ for i = 1 , . . . , n , we have the following dynamics:(4.38) ˙ Z i = A i Z i , where A i ( t ) := 1 d i (cid:18) γ ( δ + d i γ ) γ δ − γ ( δ + d i γ ) (cid:19) . Note that A i depends on i only through the d i ’s. The above implies at once thatboth ω i (0) , ω i ( T ) are non zero since, otherwise, from p i (0) = p i ( T ) = 0, we would have Z i (0) or Z i ( T ) = 0, and thus Z i ≡
0. However, this would yield c i ≡
0, contradicting(4.37).We now claim that the d i ’s are two by two distinct. Indeed, if this were not thecase, we would have d i = d i +1 for some i ≥
2. By (4.38), this implies that(4.39) Z i +1 = ω i +1 (0) ω i (0) Z i on [0 , T ] . Then, by (4.36), we have(4.40) c i +1 = ω i +1 (0) ω i (0) c i on [0 , T ] , which yields(4.41) Z T c i c i +1 dt = ω i +1 (0) ω i (0) Z T c i dt = b ω i +1 (0) ω i (0) = 0 . However, this contradicts (4.37) and thus proves the claim.Let us now denote by p and m the characteristic polynomials of the matrices P Q and M , respectively. That is, the degree n polynomials in the indeterminate ξ givenby(4.42) m ( ξ ) = det( ξ Id n − M ) , and p ( ξ ) = det( ξ Id n − P Q ) . Observe that, since P Q and Spec( M ) (taking into account multiplicities) are indepen-dent of t ∈ [0 , T ], the same is true for p and m . In order to complete the proof of thestatement, we will compute the ratio m / p in two different ways.Firstly, we observe that, by definition of M , it holds(4.43) m ( ξ ) p ( ξ ) = det (cid:0) Id n +( ξ Id n − P Q ) − ( ωω ⊤ + pω ⊤ + ωp ⊤ ) (cid:1) = det(Id n + V W ⊤ ) , where we defined the n × V and W by(4.44) V = (( ξ Id n − P Q ) − ω, ( ξ Id n − P Q ) − p ) and W = ( ω + p, ω ) . hen, Sylvester’s determinant identity yields(4.45) m ( ξ ) p ( ξ ) = det(Id + W ⊤ V ) = 1 + n X i =1 ω i + 2 ω i p i + X j = i ( ω i p j − ω j p i ) d i − d j ξ + d i . Here, in the last equality we have used the fact that the d i ’s are two by two distinct.On the other hand, by partial fraction decomposition of m / p in terms of theindeterminate ξ , we have(4.46) m ( ξ ) p ( ξ ) = 1 + n X i =1 m ( − d i ) p ′ ( − d i ) 1 ξ + d i . By comparing the above with (4.45), we finally obtain the following integrals of motion(4.47) m ( − d i ) p ′ ( − d i ) = ω i + 2 ω i p i + X j = i ( ω i p j − ω j p i ) d i − d j , on [0 , T ] . The statement then follows by evaluating the above at t = 0 and t = T , and usingthe transversality conditions p (0) = p ( T ) = 0.We are finally in a position to prove the main result of this section. Proof of Proposition 4.1.
Let λ = ( ω, Q, p, P Q ) be an extremal of (OCP) associ-ated with an optimal control S . By Proposition 4.4, up to an orthonormal change ofbasis, there exist α ∈ (0 , r ∈ N , and a positive diagonal matrix D Q ∈ R r × r suchthat we have the following dichotomy:1. r = n − Q ( T ) = diag( a, D Q ) and P Q = diag( α, n − );2. r = n − − k for some k ∈ { , . . . , n − } , Q ( T ) = diag( a, b Id k , D Q ) and P Q = diag( α, − D b , r ) for some positive diagonal matrix D b ∈ R k × k ;In the first case, by Proposition 4.5, we have J ( S, ω (0)) = a . Then, it sufficesto consider a periodic version S ∗ ∈ Sym ( P E ) n ( a, b, T ) of the control S ∈ Sym + n ( a, b, T )defined in Proposition 2.1. Indeed, it holds J ( S, ω (0)) = a , and thus S is an optimalcontrol, and moreover the corresponding trajectory is periodic since ω ≡ ω (0).Let us now focus on the second case. We start by assuming that r = 0, i.e., that λ satisfies Assumption 4.6. Then, since M S ≡ M = n − S = cc ⊤ for some vector-valued function c . Moreover, byProposition 4.8, there exists a diagonal matrix D with entries ± ω ( T ) = Dω (0) and c ( T ) = Dc (0). We next define the required S ∗ ∈ Sym ( P E ) n (2 a, b, T ) as S ∗ = c ∗ c ⊤∗ , where c ∗ is the 2 T -periodic vector valued function satisfying c ∗ ( t ) = c ( t )for t ∈ [0 , T ] and c ∗ ( t ) = Dc ( t − T ) for t ∈ [ T, T ]. Clearly, the correspondingtrajectory ω ∗ starting at ω (0) will satisfy ω ∗ ( t ) = Dω ( t − T ) for t ∈ [ T, T ] due tothe fact that D ∈ O( n ) and the invariance of the dynamics by elements of O( n ). Inparticular, ω ∗ (2 T ) = ω ∗ (0) since D = Id n and similarly c ∗ (2 T ) = c ∗ (0).Finally, the case r ≥ r = 0 as follows. Observe thatthe control diag( S ,
0) where S ∈ Sym + k +1 ( a, b, T ) is the rank-one optimal controlgiven by Proposition 4.7 with initial condition ω (0), has cost J (diag( S , , ˜ ω (0)) = µ ( a, b, k + 1) = µ ( a, b, n ) for ˜ ω (0) = ( ω (0) , S ∈ Sym + r ( a, b, T ).Define the control S ∈ Sym + n ( a, b, T ) as follows:(4.48) S ( t ) = ( S (2 t ) , , for t ∈ [0 , T / , , S (2 t − T )) , for t ∈ [ T / , T ] . his is a rank-one optimal control whose associated trajectory satisfies ω ( T ) = Dω (0)for some diagonal matrix D with entries ±
1. Then, the same procedure used in thecase r = 0 yields the desired S ∗ ∈ Sym ( P E ) n (2 a, b, T ).
5. The 2D case.
In this section, we prove the first part of Proposition 1.6,which, thanks to Proposition 2.4 reduces to the following.
Proposition
There exists an universal constant C > such that, for every < a ≤ b , one has (5.1) µ ( a, b, ≤ C a b . We start by introducing adapted notations for the 2D case. Given a vector v ∈ R we denote by v ⊥ its counter-clockwise rotation of angle π/
2. Moreover, for θ ∈ R wewrite c θ and s θ to denote cos θ and sin θ , respectively.Observe that M is equal to S × Sym and, thanks to Proposition 4.4, up to anorthonormal change of basis, we can represent an extremal λ = ( ω, Q, p, P Q ) ∈ T ∗ M as λ = ( θ, Q, η, α, d ) ∈ R × Sym × R × (0 , × [0 , + ∞ ), via the following identifications(5.2) ω = e iθ/ , p = ηω ⊥ = ηie iθ/ , P Q = diag( α, − d ) , Q ( T ) = diag( a, b ) . In this subsection, we consider a fixed extremal λ = ( θ, Q, η, α, d ) satisfying Assumption 4.6, and associated with an optimal control S = cc ⊤ of constant trace. This implies d >
0. The (PE) condition and (5.2) thenyields, up to a time reparametrization, that(5.3) k c k = 1 and T = a + b. Moreover, the matrix M defined in (4.8) is non-trivial and can be written on [0 , T ] as(5.4) M = diag( α, − d ) − η (cid:0) ω ( ω ⊥ ) ⊤ + ω ⊥ ω ⊤ (cid:1) − ωω ⊤ . Since
M c = 0 on [0 , T ] and the trace of M is constant and equal to α − d −
1, it holds(5.5) M ( t ) = ( α − d − c ⊥ ( t )( c ⊥ ( t )) T , for a.e. t ∈ [0 , T ] . Since this ensures that c is actually absolutely continuous, this equality holds on thewhole interval [0 , T ].In the following result, we rewrite the dynamics of an extremal trajectory withthe adapted notations for the 2D case. Lemma
Letting c = e iφ/ , φ ∈ R , we have the following dynamics (5.6) ˙ θ = s θ − φ , ˙ η = − s θ − φ + 2 ηc θ − φ , ˙ φ = 2 η − α + d . Moreover, η (0) = η ( T ) = 0 and, for every t ⋆ ∈ [0 , T ] such that η ( t ⋆ ) = 0 , we have c θ ( t ⋆ ) = 1 − d (1 − α ) α + d , (5.7) c φ ( t ⋆ ) = − d (1 + d )( α + d )(1 − α + d ) , (5.8) with s θ ( t ⋆ ) s φ ( t ⋆ ) < . In addition, M ( t ) c ( t ) = 0 along trajectories of (5.6) satisfyingthe previous conditions, for t ∈ [0 , T ] . roof. The transversality conditions (Item (ii) of Definition 4.2) imply immedi-ately that η (0) = η ( T ) = 0. The first two equations of (5.6) follow at once from(4.4)-(4.6) and (5.2). Let us prove the last one. Due to the fact that 2 ˙ c = ˙ φc ⊥ and2 ˙ c ⊥ = − ˙ φc , differentiating (5.5) yields(5.9) ˙ M = 1 − α + d φ ( c ⊥ c ⊤ + c ( c ⊥ ) ⊤ ) . In particular,(5.10) c ⊤ ˙ M c ⊥ = 1 − α + d φ. Hence, replacing the expression of ˙ M given in (4.33) and using (5.2), the left-handside of (5.10) turns out to be equal to η . This proves the last equation of (5.6).Equations (5.7) and (5.8) follow at once by developing the equation M ( t ⋆ ) c ( t ⋆ ) = 0at every t ⋆ ∈ [0 , T ] such that η ( t ⋆ ) = 0.We now let R φ/ be the matrix corresponding to the counter clock-wise rotationby φ/ M = R ⊤ φ/ M R φ/ . Direct computations show thatthe components ˜ M , and ˜ M , of ˜ M are given by2 ˜ M , = − α − d + ( α + d ) c φ − c θ − φ + 2 ηs θ − φ , (5.11) 2 ˜ M , = − ( α + d ) s φ − s θ − φ − ηc θ − φ . (5.12)As the first column of ˜ M is equal to zero, we have that ˜ M , = ˜ M , = 0 along thetrajectory. In particular one has 2 ˜ M , c θ − M , s θ = ( − α − d ) c θ + ( α + d ) c φ c θ +( α + d ) s φ s θ − c φ − ηs φ = 0 so that, from (5.7) and (5.8), if η ( t ⋆ ) = 0 then(5.13) s θ ( t ⋆ ) s φ ( t ⋆ ) = − αd (1 − α )(1 + d )1 − α + d < . Note that, if η ( t ⋆ ) = 0, then (5.7), (5.8) and (5.13) are actually equivalent to thecondition ˜ M , = ˜ M , = 0. Hence, to conclude the proof it is enough to showthat if this condition is satisfied at t = 0, then it is satisfied for t ∈ [0 , T ]. Thisis an immediate consequence of the fact that, differentiating ˜ M , , ˜ M , along thesystem (5.6), one obtains the following linear system(5.14) ˙˜ M , = η − α + d ˜ M , , ˙˜ M , = − η − α + d ˜ M , . Observe that, for any ( θ, η, φ ) satisfying the conditions of Lemma 5.2, the triple(2 π − θ, π − η, φ ), corresponding to a reflection of ω and q with respect to thevertical axis, also satisfies such conditions and has the same cost. Note that α < φ = 0 and the corresponding trajectory of (5.6) is constant, contradictingthe (PE) condition. Hence, without loss of generality, we assume in the sequel that φ = φ (0) ∈ (0 , π ). We next show that the dynamic of the control c is actuallyindependent of θ and η . Proposition
The control c = e iφ/ , φ ∈ R , satisfies the pendulum equation (5.15) ¨ φ = 12 ν s φ , where ν = s − α + d α + d ) , ith period T /κ for some κ ∈ N ∗ . Moreover, η ( t ) = 0 if and only if t = jT /κ , for j ∈ J , κ K , and the following relations hold (5.16) a = νκK + ( φ ) , b = νκK − ( φ ) , where K ± ( γ ) = Z πγ ± c φ √ c γ − c φ dφ. Proof.
From (5.12) and the second equation in (5.6) one obtains that ˙ η = ( α + d ) s φ . By taking the time derivative of the last equation of (5.6), we then get (5.15).Since ˙ Q = cc ⊤ , we have diag( a, b ) = Q ( T ) = R T cc ⊤ dt . By simple computations,we have that(5.17) cc ⊤ = 12 (cid:18) Id + (cid:18) c φ s φ s φ − c φ (cid:19)(cid:19) . This yields at once that(5.18) a = 12 T + Z T c φ dt ! and b = 12 T − Z T c φ dt ! . The statement follows by standard facts on the pendulum equation, see [5].It is convenient to rewrite the functions K ± in terms of classical elliptic integrals.By a simple change of coordinates one obtains K + ( φ ) = 2 √ K ( c φ / ) − E ( c φ / ) , (5.19) K − ( φ ) = 2 √ E ( c φ / ) , (5.20)where(5.21) K ( x ) := Z π/ p − x s u du, E ( x ) := Z π/ p − x s u du are the complete elliptic integrals of the first and second kind, respectively. We recallthat K, E are monotone functions such that K ( x ) ≥ E ( x ) for any x ∈ [0 ,
1) and withequality only if x = 0. Moreover one has that K (0) = E (0) = π , lim x → K ( x ) = ∞ , E (1) = 1 , (5.22) dEdx ( x ) = E ( x ) − K ( x ) x , dKdx ( x ) = E ( x ) x (1 − x ) − K ( x ) x , (5.23) lim x → + K ( x ) − E ( x ) x = π . (5.24)We show below that the conditions obtained in Lemma 5.2 completely characterize(up to orthogonal transformations of the coordinates) the extremals of (OCP). Proposition
Let ( α, d ) ∈ (0 , × (0 , ∞ ) and assume that K + ( φ ) < K − ( φ ) ,where φ ∈ (0 , π ) is defined by (5.8) . Then the solutions of (5.6) satisfying theconditions of Lemma 5.2 correspond to extremal trajectories λ = ( θ, diag( a, b ) , η, α, d ) for (OCP) , for a = νκK + ( φ ) , b = νκK − ( φ ) , for every positive integer κ .On the other hand, for any < a < b and positive integer κ , there exists (upto time-invariant orthogonal transformations of the components ( ω, Q, p ) ) a uniqueextremal trajectory for (OCP) with trace identically equal to one such that p exactlyvanishes κ + 1 times on its interval of definition [0 , a + b ] . This trajectory correspondsto a solution of (5.6) for some ( α, d ) ∈ (0 , × (0 , ∞ ) . roof. The first part of the proposition easily follows from the results provedabove. Indeed, trajectories of (5.6) satisfy ( i ) in Definition 4.2 by definition, while( iii ) follows from Lemma 5.2 and (5.5). Proposition 5.3 shows that Q ( a + b ) =diag( a, b ) ∈ Sym n ( a, b ) with a = νκK + ( φ ) < b = νκK − ( φ ) , for some positiveinteger κ . In particular P Q belongs to the normal cone of Sym n ( a, b ), proving ( ii ) inDefinition 4.2.To prove the second part of the proposition, we will establish a one-to-one cor-respondence between the pairs of positive numbers ( a, b ) and the possible pairs ofparameters ( α, d ). This is enough to conclude the proof since Lemma 5.2 identifiesa unique extremal trajectory up to a reflection, and in view of Proposition 4.4 andProposition 5.3. We first notice that the map ( ν, φ ) ( νκK + ( φ ) , νκK − ( φ )) is abijection from (0 , ∞ ) × (0 , π ) to (0 , ∞ ) × (0 , ∞ ). Indeed, by (5.22)-(5.23), one deducesthat the map φ K + ( φ ) /K − ( φ ) is strictly decreasing and, moreover,(5.25) lim φ → K + ( φ ) K − ( φ ) = ∞ , lim φ → π K + ( φ ) K − ( φ ) = 0 . We now show that the map ( α, d ) ( ν, φ ), with φ and ν satisfying (5.8) and(5.15), is also a bijection from (0 , × (0 , ∞ ) to (0 , ∞ ) × (0 , π ). For this purpose wenotice that for any φ ∈ (0 , π ) , α ∈ (0 , d . In particular it admits a unique positive zero d φ ( α ) = ( − p φ / α (1 − α )). Substituting this expression into (5.15) it iseasy to see that ν is a strictly decreasing function of α for any given φ ∈ (0 , π ), withlim α → ν = ∞ and lim α → ν = 0. It follows that for any ( ν, φ ) ∈ (0 , ∞ ) × (0 , π ) thereexists a unique pair ( α, d ) ∈ (0 , × (0 , ∞ ) such that (5.8) and (5.15) are satisfied,which concludes the proof of the proposition. Taking into account Proposition 2.1, it isenough to establish Proposition 5.1 for sequences ( a l , b l ) l ∈ N such that b l tends toinfinity as l tends to infinity. Moreover, since we need to upper bound µ ( a, b, cc ⊤ ∈ Sym +2 ( a, b, a + b ) and an initial condition ξ ∈ S , whosecost J ( cc ⊤ , ξ ) is indeed smaller than C a/ (1 + b ) for some universal constant C . Weclaim that such a control is provided by Proposition 5.4 in the case κ = 1. Showingthis claim simply amounts to compute the cost of such a control and to verify thedesired inequality. In order to do so, we introduce some preliminary estimates. Lemma
Consider the extremal trajectories described in Proposition 5.4 with κ = 1 . Then, the following hold true, (5.26) α ∼ b →∞ K − ( φ ) b s φ / , and d ∼ b →∞ K − ( φ ) b c φ / . Moreover, there exist two positive constants C , C independent of φ such that, (5.27) C ab ≤ c φ / ≤ C ab , and there also exist ¯ C , ¯ C > and b > such that (5.28) ¯ C ab ≤ d √ α ≤ ¯ C ab , ∀ < a ≤ b s.t. b > b . Proof.
To prove (5.26) we notice that the function(5.29) F : ( α, d ) (cid:16) ν , c φ ν (cid:17) = (cid:16) K − ( φ ) b , (1 + c φ ) K − ( φ ) b (cid:17) , here c φ and ν are given by (5.8) and (5.15), maps the origin to itself and, as itsdifferential at the origin is given by(5.30) DF (0) = (cid:18) (cid:19) , one can apply the inverse function theorem in a neighborhood O of the origin. Since b goes to infinity if and only if ν goes to infinity, and this implies that α, d go to zeroas well as the value of F , we can write(5.31) ( α, d ) ∼ b →∞ DF (0) − (cid:16) K − ( φ ) b , (1 + c φ ) K − ( φ ) b (cid:17) which proves (5.26).To get (5.27), first notice that φ verifies the constraint K + ( φ ) ≤ K − ( φ ) imply-ing by (5.19) and (5.20) and the properties of E and K that x = c φ / must belong toan interval [0 , x ∗ ] with x ∗ <
1. On the other hand, by again using (5.19) and (5.20),it follows that proving (5.27) is equivalent to show positive lower and upper boundsfor the function x x E ( x ) K ( x ) − E ( x ) with x ∈ [0 , x ∗ ]. It is then enough to prove that theprevious function admits positive limits as x tends to zero and x ∗ , which clearly holdstrue by (5.24) and the fact that x ∗ < b large enough and using (5.26), one gets that there exists b > b > b it holds(5.32) 12 K − ( φ ) c φ / √ bs φ / ≤ d √ α ≤ K − ( φ ) c φ / √ bs φ / . By a reasoning similar to the one yielding (5.27) and using the fact that K − ( φ ) s φ / isuniformly bounded by two positive constants in the admissible range, one deduces(5.28). Lemma
Consider the extremal trajectories described in Proposition 5.4 with κ = 1 . Then, there exist b > and two positive constants C , C such that (5.33) C ab ≤ J ( cc ⊤ , ω ) ≤ C ab , ∀ b > b , a ≤ b. Proof.
From (OCP), one has at once that(5.34) J ( cc ⊤ , ω ) = Z a + b c θ − φ ) / dt, and c θ − φ ) / = s ε/ , where ε = θ − φ − π . From (5.6), the dynamics of ε on [0 , a + b ]is given by(5.35) ¨ ε = − µs ε + s ε c ε , where µ = 11 − α + d . Moreover, the initial conditions ( ε , ˙ ε ) = ( ε (0) , ˙ ε (0)) satisfy the relations(5.36) c ε = 1 − αd − α + d , ˙ ε = − s ε = − ( α + d ) s φ , and ( ε ( a + b ) , ˙ ε ( a + b )) = ( − ε , − ˙ ε ). Notice also that if there exists t ∈ [0 , a + b ]such that ε ( t ) = 0, then ε ( t + t ) = − ε ( t − t ) for times t − t, t + t in [0 , a + b ]. e have the following first integral for ε after integrating between the times zeroand t ∈ [0 , a + b ] and taking into account (5.36):(5.37) ˙ ε = 2 µ ( c ε − c ε ) + s ε . Since ε starts at time t = 0 with negative speed ˙ ε according to (5.36), ε will decreasein a right neighborhood of t = 0. Note that, from (5.37), ˙ ε will keep the same sign,i.e., negative, as long as | ε | ≤ ε . Hence, ε will reach the value ε = − ε at a time t ,however with a negative speed. Therefore, by (5.36), t must be strictly smaller than a + b and ε decreases in a right neighborhood of t = t . This will go on till either ˙ ε = 0or ε = − π/
2, since at time t = a + b we have ε ( a + b ) = − ε and ˙ ε ( a + b ) = − ˙ ε . Thelatter possibility is clearly ruled out since the r.h.s. of (5.37) is negative at ε = − π/ b sufficiently large. Then, ˙ ε = 0 occurs at some time ¯ t < a + b for ε = − ¯ ε , where¯ ε is the unique angle in (0 , π/
2) verifying(5.38) 2 µ ( c ¯ ε − c ε ) + s ε = 0 . Since ε (¯ t ) is a minimum for ε , one must necessarily have that ¨ ε (¯ t ) ≥
0. On the otherhand, (5.35) can be written ¨ ε = − s ε ( µ − c ε ), yielding that µ − c ε (¯ t ) ≥ s ε (¯ t ) < µ − c ¯ ε ) = 1 + µ − µc ε = ( α + d ) (1 − α + d ) , which is strictly positive. Then ¯ t is an isolated zero of ˙ ε and the latter must changesign there, implying that ε increases in a right neighborhood of t = ¯ t . By a similarreasoning as before, ε increases till ε = − ε at a time τ and one will also get that˙ ε ( τ ) = − ˙ ε = − s ε ( τ ) .We next show that τ = a + b . Notice first that ε is periodic of period equal to 2 τ and moreover there must exists an integer m such that a + b = (2 m + 1) τ . Since ε satisfies the equation(5.40) ˙ ε = − ε − η − α + d one deduces that η is periodic with period less than or equal to the one of ε . Finally,recall that the minimal period of η coincides with that of φ , which is equal to 2( a + b )since ν = 1. Hence 2(2 m + 1) τ = 2( a + b ) ≤ τ implying that τ is equal to a + b .To provide an estimate to (5.34) let us first derive an asymptotics for ¯ ε as b tendsto infinity. From (5.39), (5.35) and the fact that µ ≥ c ε (¯ t ) , we have(5.41) c ¯ ε = µ − p µ − µc ε = 1 − d − α + d , which yields(5.42) ¯ ε ∼ √ d, as b → + ∞ . By the previous claim, | ε | ≤ √ d for b large enough on [0 , a + b ]. Subtracting(5.38) to (5.37) yields(5.43) ˙ ε = 2 µ ( c ε − c ¯ ε ) − c ε + c ε = ( c ε − c ¯ ε ) (cid:0) ( µ − c ε ) + ( µ − c ¯ ε ) (cid:1) . e have µ − c ¯ ε = ( α + d ) (cid:0) O ( α ) (cid:1) and µ − c ε = ( α − d ) (cid:0) O ( α ) (cid:1) + 2 s ε/ . Hence(5.44) ( µ − c ¯ ε ) + ( µ − c ε ) = (2 α + ε / (cid:0) O ( α ) (cid:1) . On the other hand,(5.45) c ε − c ¯ ε = ¯ ε − ε (cid:0) O ( α ) (cid:1) . Gathering the previous inequalities then yields(5.46) ˙ ε = (¯ ε − ε )( α + ( ε/ )(1 + O ( α )) , where O ( α ) denotes a function of the time such that | O ( α ) | ≤ cα on the interval[0 , a + b ], for some c > b .We can finally prove the desired estimate for the cost. Indeed, by taking intoaccount the previous results, one has(5.47) J ( cc ⊤ , ω ) = 2 Z ¯ ε s ε/ dε (cid:0) µ ( c ε − c ¯ ε ) − c ε + c ε (cid:1) / . Using (5.46), the above equation can be rewritten as(5.48) J ( cc ⊤ , ω ) ∼ (1 + O ( α ))2 Z ¯ ε ε dε (cid:0) (¯ ε − ε )( α + ε ) (cid:1) / , as b → + ∞ . Thanks to (5.42), this further simplifies to J ( cc ⊤ , ω ) ∼ (1 + O ( α )) d √ α J ( d/α ), where(5.49) J ( γ ) = 2 Z v dv (cid:0) (1 − v )(1 + 2 γv ) (cid:1) / , γ > . Since d/α is bounded (this comes from (5.26) and the fact that c φ / belongs to aninterval [0 , x ∗ ] with x ∗ < J ( d/α ) is bounded below and above by positive constantsindependent of ( a, b ). Together with the inequalities (5.28), this concludes the proofof the statement. Appendix A. A weak- ∗ compactness result. For
T >
0, let L ∞ denote the space of essentially bounded real symmetric matrix-valued functions M : [0 , T ] → R n × n . As any two norms are equivalent in finitedimensional spaces, the definition of L ∞ is independent of the choice of the norm inthe vector subspace of R n × n made of real symmetric matrices. The space L ∞ canbe identified with the dual of the space of integrable real symmetric matrix-valuedfunctions L , via the duality(A.1) h M, A i = Z T Tr( M ( t ) A ( t )) dt, M ∈ L ∞ , A ∈ L . We recall that a sequence ( M ℓ ) ℓ ≥ ⊂ L ∞ is weakly- ∗ convergent to M ∈ L ∞ if(A.2) lim ℓ → + ∞ h M ℓ , A i = h M, A i , ∀ A ∈ L . emma A.1.
The set (A.3) S = { S ∈ Sym + n ( a, b, T ) | Tr S is constant for a.e. t ∈ [0 , T ] } , is weakly- ∗ compact in L ∞ .Proof. Notice that for any S ∈ S , we have for a.e. τ ∈ [0 , T ] that(A.4) k S ( τ ) k = Tr( S ( τ )) = 1 T Z T Tr S ( t ) dt ≤ bnT , where k A k := Tr √ A T A is the Schatten 1-norm of a matrix A ∈ R n × n . Indeedthe first equality comes from the fact that S takes positive semi-definite values, thesecond one from the definition of S ∈ S , and the last inequality from Condition(INT). This shows that S is a bounded subset of L ∞ . By Banach-Alaoglu theorem,it then suffices to prove that S is weak- ∗ closed in L ∞ . To this effect, let ( S ℓ ) ℓ ≥ asequence in S which weakly- ∗ converges to S ∈ L ∞ . For ℓ ≥
0, let T ℓ be the constantvalue taken by the function t Tr S ℓ ( t ). Since T ℓ = h S ℓ , Id n i /T for l ≥
0, it followsthat lim ℓ →∞ T ℓ = h S, Id n i /T .For every ε ∈ (0 , T ) and τ ∈ [ ε, T − ε ], let χ τε : [0 , T ] → { , } be the characteristicfunction of the interval [ τ − ε, τ + ε ]. Then, by Lebesgue theorem and the definitionof weak- ∗ convergence, it holds for a.e. τ ∈ [0 , T ](A.5) Tr S ( τ ) = lim ε ↓ ε h S, χ τε Id n i = lim ε ↓ ε lim ℓ → + ∞ h S ℓ , χ τε Id n i . Since S ℓ ∈ S we have that h S ℓ , χ τε Id n i = 2 ε T ℓ if τ ∈ [ ε, T − ε ], which finally yieldsthat, for a.e. in τ ∈ [0 , T ], Tr S ( τ ) is equal to the constant h S, Id n i /T .We are left to show that S ∈ Sym + n ( a, b, T ). We have that(A.6) a k x k ≤ x ⊤ Z T S ℓ ( t ) dt ! x = h S ℓ , xx ⊤ i ≤ b k x k for any x ∈ R n and ℓ ≥
0, and passing to the limit as ℓ goes to infinity we obtain a Id n ≤ R T S ( t ) dt ≤ b Id n . Moreover, again by Lebesgue theorem, for a.e. τ ∈ [0 , T ]and x ∈ R n we have(A.7) x ⊤ S ( τ ) x = Tr( S ( τ ) xx ⊤ ) = lim ε ↓ ε lim ℓ → + ∞ h S ℓ , χ τε xx ⊤ i . Since h S ℓ , χ τε xx ⊤ i = R T χ τε ( t ) x ⊤ S ℓ ( t ) xdt ≥
0, we obtain that x ⊤ S ( τ ) x ≥ τ ∈ [0 , T ]. This completes the proof of the lemma. Appendix B. Hamiltonian equations.
In this appendix, we apply the Pontryagin Maximum Principle (PMP) to thecontrol system (OCP), in order to derive necessary optimality conditions. These areessential to the proofs of Proposition 4.3.For the Hamiltonian formalism used below, we refer to [1].
Lemma
B.1.
Let H ∈ C ∞ ( T ∗ S n − ) be an Hamiltonian function. Upon the identi-fication T ∗ ω S n − ≃ ω ⊥ , the corresponding Hamiltonian system ˙ ξ = ~H ( ξ ) , ξ = ( ω, p ) ∈ T ∗ S n − , reads ˙ ω = ∂H∂p − (cid:18) ω ⊤ ∂H∂p (cid:19) ω (B.1) ˙ p = ∂H∂ω − (cid:18) ω ⊤ ∂H∂ω (cid:19) ω − (cid:18) ω ⊤ ∂H∂p (cid:19) p + (cid:18) p ⊤ ∂H∂p (cid:19) ω. (B.2) roof. Upon the given identifications, we have that(B.3) T ( ω,p ) ( T ∗ S n − ) = (cid:8) ( v , v ) ∈ R n | ω ⊤ v = 0 and p ⊤ v + ω ⊤ v = 0 (cid:9) . Letting ( ∂H∂ω , ∂H∂p ) ∈ R n be the partial derivative at ξ = ( ω, p ) ∈ S n − of H , we have(B.4) d ξ H ( v , v ) = v ⊤ ∂H∂ω + v ⊤ ∂H∂p , ∀ ( v , v ) ∈ T ( ω,p ) ( T ∗ S n − ) . On the other hand, the Hamiltonian vector field ~H ∈ Γ( T ∗ S n − ), with components ~H = ( ~H p , − ~H ω ) ∈ T ( T ∗ S n − ), is the only vector field such that(B.5) d ξ H ( v , v ) = v ⊤ ~H ω ( ξ ) + v ⊤ ~H p ( ξ ) , ∀ ( v , v ) ∈ T ( ω,p ) ( T ∗ S n − ) . As a consequence of these two facts, we have ω ⊤ ~H p ( ω, p ) = 0 , p ⊤ ~H p ( ω, p ) = ω ⊤ ~H ω ( ω, p ) , (B.6) v ⊤ (cid:18) ∂H∂ω − ~H ω ( ω, p ) (cid:19) + v ⊤ (cid:18) ∂H∂p − ~H p ( ω, p ) (cid:19) = 0 , ∀ ( v , v ) ∈ T ( ω,p ) ( T ∗ S n − ) . (B.7)By (B.3), we have that, if ( v , v ) ∈ T ( ω,p ) ( T ∗ S n − ) is such that ω ⊤ v = 0, then v = 0. As a consequence, considering (B.7) for such ( v , v ) and taking into accountthe first equation of (B.6), yields(B.8) ~H p ( ω, p ) = ∂H∂p − (cid:18) ω ⊤ ∂H∂p (cid:19) ω. Plugging this in (B.7), one deduces that(B.9) ~H ω ( ω, p ) = ∂H∂ω − (cid:18) ω ⊤ ∂H∂ω (cid:19) ω − (cid:18) ω ⊤ ∂H∂p (cid:19) p + (cid:18) p ⊤ ∂H∂p (cid:19) ω. This completes the proof.
Proposition
B.2.
Let ( ω, Q ) : [0 , T ] → M be an optimal trajectory of system (OCP) , associated with an optimal control S . Then, there exists a curve t ∈ [0 , T ] ( p ( t ) , P Q ( t )) ∈ T ∗ ω ( t ) S n − × T ∗ Q ( t ) Sym n and ν ∈ { , } , satisfying (4.9) , (4.10) , (4.11) ,and (4.13) .Proof. Let λ = ( ω, Q, p, P Q ). Recall that we consider the identification T ∗ ω S n − × T ∗ Q Sym n ≃ ( R ω ) ⊥ × Sym n . We also let T ( ω,Q ) M ≃ ( R ω ) ⊥ × Sym n , so that h λ, v i = p ⊤ v + Tr( P Q V ) = Tr( vp ⊤ + V P Q ) , for every v = ( v, V ) ∈ T ( ω,Q ) M . We follow the formulation of the PMP given in [1, Theorem 12.4]. Simple computa-tions show that, for ν ∈ { , } , the Hamiltonian associated with the system is givenby (up to constants)(B.10) H ( λ, S ) = Tr( S ˜ M )2 , where ˜ M = P Q − (cid:0) ωp ⊤ + pω ⊤ + ν ωω ⊤ (cid:1) . Equations (4.11) and (4.13) are immediate consequences of the PMP. In order tocomplete the proof, we are hence left to check (4.9) and (4.10). By the PMP, we have λ = ~H , where we let ~H ∈ Γ( T ∗ M ) be the Hamiltonian vector field associated withthe control S .We observe that the Hamiltonian decomposes as H ( ω, Q, p, P Q , S ) = H ( ω, p, S )+ H ( Q, P Q , S ). This implies that a similar decomposition holds for the correspondingHamiltonian vector field. Thus, (4.9) follows by Lemma B.1 and the fact that(B.11) ∂H ∂ω = − Sp − ν Sω, and ∂H ∂p = − Sω.
On the other hand, (4.10) follows from the easily verified fact that(B.12) ~H = (cid:18) ∂H ∂P Q , − ∂H ∂Q (cid:19) = ( S, . This concludes the proof of the statement.
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