Large Deviations for the Reliability Assessment of Redundant Multi-Channel Systems
aa r X i v : . [ m a t h . D S ] M a r Large Deviations for the Reliability Assessmentof Redundant Multi-Channel Systems
Getachew K. Befekadu and Panos J. Antsaklis
Abstract
In this paper, we are concerned with the reliability assessment of redundant multi-channel systemshaving multiple controllers with overlapping functionality – where all controllers are required to respondoptimally to the non-faulty controllers so as to ensure or maintain some system properties. In particular,for such redundant systems with small random perturbation, we study the relationships between theexit probabilities with which the state-trajectories exit from a given bounded open domain and thevalue functions corresponding to a family of stochastic exit-time control problems on the boundary ofthe given domain. Moreover, as the random perturbation vanishes, such relationships provide usefulinformation concerning the reliability of the redundant multi-channel systems arising from the largedeviations problem in connection with the asymptotic estimates of exit probabilities with respect tosome portions of the boundary of the given domain. Finally, we briefly comment on the implication ofour results on a co-design technique using multi-objective optimization frameworks for evaluating theperformance of the redundant multi-channel systems.
Index Terms
Boundary exit problem, large deviations, reliable system, redundant multi-channel system, smallrandom perturbations.
G. K. Befekadu is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA.E-mail: [email protected]. J. Antsaklis is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA.E-mail: [email protected]
I. I
NTRODUCTION
In this paper, we are concerned with the reliability assessment of redundant multi-channelsystems having multiple controllers with overlapping functionality. Specifically, we considera redundant system with multi-controller configurations – where all controllers are requiredto respond optimally, in the sense of best-response correspondence, i.e., a reliable-by-design requirement, to the non-faulty controllers so as to ensure or maintain some system properties.Here we are mainly interested in a systematic understanding of the relationships between theexit probabilities with which the state-trajectories exit from a given bounded open domainand the value functions corresponding to a family of stochastic exit-time control problems onthe boundary of the given domain. As a consequence of such relationships, we obtain usefulinformation concerning the reliability of the redundant multi-channel systems arising from thelarge deviations problem as the random perturbation vanishes in connection with the asymptoticestimates of exit probabilities with respect to some portions of the boundary of the given domain.Moreover, we also comment on the implication of our results on a co-design technique usingmulti-objective optimization frameworks for either evaluating the performance or finding anappropriate set of redundant controllers for the multi-channel systems with respect to someprescribed portions of the boundary of the given domain.It is worth mentioning that some interesting studies on the exit probabilities for the dynamicalsystems with small random perturbation have been reported in literature (see, e.g., [1], [2], [3]and [4] in the context of large deviations; see [5], [6], [7] and [8] in connection with optimalstochastic control problems; and see [5] or [9] via asymptotic expansions approach). Note that therationale behind our framework follows, in some sense, the settings of these papers – where weestablish a connection between the asymptotic estimates of the exit probabilities and the stochasticexit-time control problems on some portions of the boundary of the given domain. However,to our knowledge, such a connection has not been addressed in the context of multi-channelsystems with multi-controller configurations having “overlapping or backing-up” functionality,and it is important because it provides a framework that shows how the asymptotic estimateson the exit probabilities can be systematically used to obtain useful information concerning the reliability of the redundant multi-channel systems. The rest of the paper is organized as follows. In Section II, we present some preliminary resultsthat are useful for our main results. In Section III, we briefly discuss a family of boundaryvalue problems for the multi-channel system in the presence of small random perturbations. InSection IV, we provide asymptotic estimates on the exit probabilities on the positions of thestate-trajectories at the first time of their exit through some portions of the boundary of the givendomain. This section also provides connections between such asymptotic estimates and the valuefunctions corresponding to a family of stochastic exit-time control problems on the boundary ofthe given domain. Moreover, we use such asymptotic estimates to obtain useful information onthe reliability of the redundant multi-channel systems. Finally, Section V provides some furtherremarks. II. P
RELIMINARIES
Consider the following continuous-time multi-channel system ˙ x ( t ) = Ax ( t ) + X ni =1 B i u i ( t ) , x (0) = x , (1)where A ∈ R d × d , B i ∈ R d × r i , x ( t ) ∈ R d is the state of the system, u i ( t ) ∈ R r i is the controlinput to the i th - channel in the system.In what follows, we consider a particular class of stabilizing state-feedbacks that satisfies K ⊆ (cid:26)(cid:0) K , K , . . . , K n (cid:1)| {z } , K ∈ Y ni =1 R r i × d (cid:12)(cid:12)(cid:12)(cid:12) Sp (cid:16) A + X ni =1 B i K i (cid:17) ⊂ C − Sp (cid:16) A + X i = j B i K i (cid:17) ⊂ C − , j = 1 , , . . . , n ) , (2) Remark 1:
We remark that the above class of state-feedbacks is useful for maintaining thestability of the closed-loop system both when all of the controllers work together, i.e., (cid:0) A + P ni =1 B i K i (cid:1) , as well as when there is a single-channel controller failure in the system, i.e., In this paper, our intent is to provide a theoretical result, rather than considering a specific numerical problem or application. Sp( A ) denotes the spectrum of a matrix A ∈ R d × d , i.e., Sp( A ) = (cid:8) s ∈ C | rank( A − sI ) < d (cid:9) . (cid:0) A + P i = j B i K i (cid:1) for j ∈ { , , . . . , n } . Moreover, such a class of state-feedbacks falls withinthe redundant/passive fault tolerant multi-controller configurations with overlapping functionality(see, e.g., [10] or [11] for such a reliable-by-design requirement in multi-channel systems).Consider the following family of stochastic differential equations dx ǫ, ( t ) = (cid:16) A + X ni =1 B i K i (cid:17) x ǫ, ( t ) dt + √ ǫ σ (cid:0) x ǫ, ( t ) (cid:1) dW ( t ) , x ǫ, (0) = x (3)and dx ǫ,j ( t ) = (cid:16) A + X i = j B i K i (cid:17) x ǫ,j ( t ) dt + √ ǫ σ (cid:0) x ǫ,j ( t ) (cid:1) dW ( t ) , x ǫ,j (0) = x , j = 1 , , . . . , n, (4)where σ ∈ R d × d is a diffusion term, W (with W (0) = 0 ) is a d -dimensional Wiener process and ǫ is a small positive number, which represents the level of perturbation in the system.Let Ω ⊂ R d be a bounded open domain with smooth boundary (i.e., ∂ Ω is a manifold of class C ). Then, the second-order elliptic differential operators L ( ǫ,j ) that correspond to the abovefamily of stochastic differential systems are given by L ( ǫ, f ( j ) ( x ) = ǫ d X i,k =1 a ik ( x ) ∂ f ( j ) ( x ) ∂x i ∂x k + D(cid:16) A + X ni =1 B i K i (cid:17) x, ▽ f ( j ) ( x ) E (5)and L ( ǫ,j ) f ( j ) ( x ) = ǫ d X i,k =1 a ik ( x ) ∂ f ( j ) ( x ) ∂x i ∂x k + D(cid:16) A + X i = j B i K i (cid:17) x, ▽ f ( j ) ( x ) E , j = 1 , , . . . , n, (6)where f ( j ) ( x ) ∈ C (Ω) ∩ C ( ¯Ω) and ▽ f ( j ) ( x ) is the gradient of f ( j ) ( x ) . Further, we assumethat the matrix a ( x ) = σ ( x ) σ T ( x ) is nonnegative definite and σ ( x ) satisfies a global Lipschitzcondition. Hence, the operators L ( ǫ,j ) are uniformly elliptic for fixed ǫ > .Let C (cid:0) [0 , ∞ ) , R d (cid:1) denote the space of continuous functions from [0 , ∞ ) to R d , and let H [0 , T ] be the space of all ϕ ∈ C (cid:0) [0 , ∞ ) , R d (cid:1) such that ϕ ( t ) is absolutely continuous and R T | ˙ ϕ ( t ) | dt < ∞ for each T > . Let us associate portions of the boundary Γ j ⊂ ∂ Ω for j = 0 , , . . . , n ,with different operating conditions of the redundant system (for example, Γ with the nominaloperating condition and Γ j for j = 1 , , . . . , n , with any of single-channel failures in the system).Then, as ǫ → , we investigate the behavior of the solutions for the second-order elliptic equations corresponding to a family of boundary value problems (see equation (12)) with respect to thoseportions of the boundary Γ j ⊂ ∂ Ω for j = 0 , , . . . , n (see [12] for discussions on the firstboundary value problem with small parameter). In general, such asymptotic estimates involvefinding a family of minimum functionals I j (cid:0) ϕ, τ j (cid:1) , i.e., I j (cid:0) ϕ, τ j (cid:1) = inf ϕ ∈H [0 ,T ] ,τ j ≥ Z τ j ∧ T L j (cid:0) ϕ ( t ) , ˙ ϕ ( t ) (cid:1) dt, j = 0 , , . . . , n, (7)where the infimum is taken among all ϕ ∈ H [0 , T ] and τ j ≥ (where τ j is the first exit-timeof x ǫ,j ( t ) from Ω ) such that ϕ (0) = x and ϕ ( t ) ∈ ¯Ω for t ∈ [0 , τ j ∧ T ] , with L (cid:0) ϕ ( t ) , ˙ ϕ ( t ) (cid:1) = 12 (cid:13)(cid:13)(cid:13)(cid:13) ˙ ϕ ( t ) − (cid:16) A + X ni =1 B i K i (cid:17) ϕ ( t ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:2) a ( ϕ ( t )) (cid:3) − and L j (cid:0) ϕ ( t ) , ˙ ϕ ( t ) (cid:1) = 12 (cid:13)(cid:13)(cid:13)(cid:13) ˙ ϕ ( t ) − (cid:16) A + X i = j B i K i (cid:17) ϕ ( t ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:2) a ( ϕ ( t )) (cid:3) − , j = 1 , , . . . , n, where a ( ϕ ( t )) = σ ( ϕ ( t )) σ T ( ϕ ( t )) . Furthermore, if we let ϕ (0) = x in the domain Ω , then, for any Γ j ⊂ ∂ Ω , the infimum in (7),when subjected to an additional condition ϕ ( τ j ) ∈ Γ j , will attain I j (cid:0) x , Γ j (cid:1) = − lim ǫ → ǫ log P ( ǫ,j ) x (cid:16) x ǫ,j ( τ j ) ∈ Γ j (cid:17) , j = 0 , , . . . , n, (8)which implicitly depends on the initial condition x and the boundary Γ j . Note that suchinformation, which is based on (8), will be used to identify the exit positions on the boundary Γ j ⊂ ∂ Ω under additional assumptions on the behavior of the state-trajectories of the unperturbedsystems x ǫ,j , when ǫ = 0 , as t → ∞ .In Sections III and IV, we establish relationships between the exit probabilities with which thestate-trajectories exit from a given bounded open domain and the value functions correspondingto a family of stochastic exit-time control problems on the boundary of the given domain. Morespecifically, we provide asymptotic estimates for the exit probabilities on the positions of thestate-trajectories x ǫ,j ( t ) , for each j = 0 , , . . . , n , at the first time of their exit from a boundedopen domain Ω ⊂ R d (i.e., estimating bounds on the exit probabilities of the state-trajectories τ j = inf (cid:8) t | x ǫ,j ( t ) / ∈ Ω (cid:9) . k x k P , x T P x, x ∈ Ω . x ǫ,j ( t ) from the given domain Ω through a portion or section of the given boundary Γ j ⊂ ∂ Ω (seeProposition 1)). Such asymptotic estimates (i.e., the asymptotic estimates on the exit probabilities P ( ǫ,j ) x (cid:0) x ǫ,j ( τ j ) ∈ Γ j (cid:1) , as ǫ → , conditioned on the initial point x ∈ Ω ) can be linked to findingprobabilities for the state-trajectories x ǫ,j ( t ) that do not deviate by more than δ from a smoothfunction ϕ ∈ H [0 , T ] during the time t ∈ [0 , τ j ∧ T ] . Moreover, for small δ > , the exitprobabilities P ( ǫ,j ) x (cid:0) x ǫ,j ( τ j ) ∈ Γ j (cid:1) will have forms exp (cid:0) − ǫ I j ( x , Γ j ) (cid:1) , where I j ( x , Γ j ) is anon-negative functional of ϕ ∈ H [0 , T ] (see Proposition 2).III. B OUNDARY VALUE PROBLEM
For the family of stochastic differential equations in (3) and (4), consider the following familyof boundary value problems L ( ǫ,j ) f ( j ) ( x ) = 0 in Ω f ( j ) ( x ) = E ( ǫ,j ) x (cid:16) exp (cid:16) − ǫ Φ j (cid:0) x (cid:1)(cid:17)(cid:17) on ∂ Ω j = 0 , , . . . , n (9)where Φ j is class C function, with Φ j ≥ . Then, there exists a set of unique solutions f ( j ) ( x ) ∈ C (Ω) ∩ C ( ¯Ω) such that f ( j ) ( x ) = E ( ǫ,j ) x (cid:16) exp (cid:16) − ǫ Φ j (cid:0) x ǫ,j ( τ j ) (cid:1)(cid:17)(cid:17) , (10)where τ j is the exit-time of x ǫ,j ( t ) from the domain Ω . Note that if we further introducethe following logarithmic transformation (see, e.g., [6] or [7] for such logarithmic connectionsbetween large deviations and stochastic optimization problems) J ( ǫ,j )Φ ( x ) = − ǫ log (cid:16) f ( j ) ( x ) (cid:17) , = − ǫ log (cid:16) E ( ǫ,j ) x (cid:16) exp (cid:16) − ǫ Φ j (cid:0) x ǫ,j ( τ j ) (cid:1)(cid:17)(cid:17)(cid:17) , j = 0 , , . . . , n. (11)Then, J ( ǫ,j )Φ ( x ) satisfies the following second-order elliptic differential equation ǫ d X i,k =1 a ik ( x ) ∂ J ( ǫ,j )Φ ( x ) ∂x i ∂x k + H j (cid:0) x, ▽ J ( ǫ,j )Φ ( x ) (cid:1) in Ω ,j = 0 , , . . . , n, (12) Note that the behavior of − ǫ log P ( ǫ,j ) x (cid:0) x ǫ,j ( τ j ) ∈ Γ j (cid:1) , as ǫ → , is defined by the large deviations of the state-trajectoriesfrom their typical behavior. where H (cid:0) x, ▽ J ( ǫ, ( x ) (cid:1) = 12 (cid:13)(cid:13)(cid:13)(cid:13) ▽ J ( ǫ, ( x ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:2) a ( ϕ ( t )) (cid:3) − + (cid:28) ▽ J ( ǫ, ( x ) , (cid:0) A + X ni =1 B i K i (cid:1) x (cid:29) and H j (cid:0) x, ▽ J ( ǫ,j )Φ ( x ) (cid:1) = 12 (cid:13)(cid:13)(cid:13)(cid:13) ▽ J ( ǫ,j )Φ ( x ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:2) a ( ϕ ( t )) (cid:3) − + (cid:28) ▽ J ( ǫ,j )Φ ( x ) , (cid:0) A + X i = j B i K i (cid:1) x (cid:29) ,j = 1 , , . . . , n. Further, note that there is a duality between H j (cid:0) x, · (cid:1) and L j (cid:0) x, · (cid:1) , for each j = 0 , , . . . , n ,such that H j (cid:0) x, ▽ J ( ǫ,j )Φ ( x ) (cid:1) = inf υ (cid:26) L j (cid:0) x, ▽ J ( ǫ,j )Φ ( x ) (cid:1) + D ▽ J ( ǫ,j )Φ ( x ) , υ E(cid:27) . (13)Hence, it is easy to see that J ( ǫ,j )Φ ( x ) is a solution in class C (Ω) ∩ C ( ¯Ω) , with J ( ǫ,j )Φ = Φ j on ∂ Ω , to the dynamic programming in (12), where the latter is associated with the followingstochastic exit-time control problem J ( ǫ,j )Φ ( x , υ ( j ) ) = E ( ǫ,j ) x (cid:26)Z τ j ∧ T L j (cid:0) η ( j ) ( t ) , υ ( j ) ( t ) (cid:1) dt + Φ j (cid:0) η ( j ) ( τ j ) (cid:1)(cid:27) , j = 0 , , . . . , n, (14)and η ( j ) ( t ) satisfies the following stochastic differential equation dη ( j ) ( t ) = υ ( j ) ( t ) dt + √ ǫ σ (cid:0) η ( j ) ( t ) (cid:1) dW ( t ) (15)for j = 0 , , . . . , n (see, e.g., [13]).In the following section, i.e., Section IV, we exploit this formalism to prove the asymptoticbounds (cf. Proposition 1) for the exit probabilities on the position of state-trajectories at thefirst time of their exit from the given portion or section of the boundary of the domain Ω (cf.Proposition 2). IV. M AIN RESULTS
In this section, we present our main results, i.e., the asymptotic estimates bounds on the exitprobabilities of the state-trajectories x ǫ,j ( t ) from the given domain Ω through the given portion (or section) of the boundary Γ j ⊂ ∂ Ω for j = 0 , , . . . , n . Further, for Γ j ⊂ ∂ Ω for j = 0 , , . . . , n ,and x ∈ Ω , let q ( ǫ,j ) (cid:0) x , Γ j (cid:1) = P ( ǫ,j ) x (cid:0) x ǫ,j ( τ j ) ∈ Γ j (cid:1) , (16)and I j (cid:0) x , Γ j (cid:1) = − lim ǫ → ǫ log P ( ǫ,j ) x (cid:0) x ǫ,j ( τ j ) ∈ Γ j (cid:1) , (17)where τ j is the first exit-time of x ǫ,j ( t ) from the domain Ω . Moreover, let I j (cid:0) ϕ, τ j (cid:1) = inf ϕ ∈H [0 ,T ] ,τ j ≥ Z τ j ∧ T L j (cid:0) ϕ ( t ) , ˙ ϕ ( t ) (cid:1) dt, j = 0 , , . . . , n, (18)where the infimum is taken among all ϕ ∈ H [0 , T ] and τ j ≥ such that ϕ (0) = x , ϕ ( t ) ∈ ¯Ω for t ∈ [0 , τ j ∧ T ] and ϕ ( τ j ) ∈ Γ j . Then, we have I j (cid:0) x , Γ j (cid:1) = I j (cid:0) x , ¯Γ j (cid:1) , j = 0 , , . . . , n. (19) Remark 2:
Note that the functional Z τ j ∧ T L j (cid:0) ϕ ( t ) , ˙ ϕ ( t ) (cid:1) dt, j = 0 , , . . . , n, is lower semicontinuous with respect to ϕ and τ j ∧ T . Furthermore, the set level Ψ = (cid:8) ϕ ∈H [0 , T ] (cid:12)(cid:12) I j ( ϕ, τ j ) ≤ α (cid:9) is a compact subset of H [0 , T ] for every α ≥ and ϕ (0) = x ∈ Ω .Hence, the infimum in (18) attains a minimum on Ψ (see, e.g., [14, pp. 332, Corollary 1.4] or[7]).Next, we introduce the following assumption about the domain Ω , which is useful in the sequel. Assumption 1: If ϕ ∈ H [0 , T ] and ϕ ( t ) ∈ ¯Ω for all t ≥ , then R T L j (cid:0) ϕ ( t ) , ˙ ϕ ( t ) (cid:1) dt → + ∞ as T → ∞ for each j = 0 , , . . . , n .Consider again the stochastic control problem in (14) (together with equation (15)). Suppose that Φ M (with Φ M ≥ ) is class C such that Φ M ( x ) → + ∞ as M → ∞ uniformly on any compactsubset of Ω \ ¯Γ and Φ M ( x ) on ¯Γ j for j = 0 , , . . . , n . Further, if we let J ( ǫ,j )Φ ( x ) = J ( ǫ,j )Φ M ( x ) ,when Φ j = Φ M , then we have the following lemma. Lemma 1:
Suppose that Assumption 1 holds, then we have lim inf M →∞ x → x J ( ǫ,j )Φ M ( x ) ≥ I j (cid:0) x , ¯Γ j (cid:1) , j = 0 , , . . . , n. (20)Let Γ ◦ j denote the interior of Γ j relative to ∂ Ω and let ¯Γ j = ¯Γ ◦ j . Then, we have the followingproposition. Proposition 1:
Suppose that Assumption 1 holds, then, for j = 0 , , . . . , n , we have ǫ log P ( ǫ,j ) x (cid:0) x ǫ,j ( τ j ) ∈ Γ j (cid:1) → I j (cid:0) x , Γ j (cid:1) as ǫ → (21)uniformly for all x in any compact subset Λ ⊂ Ω . Proof : For any fixed j ∈ { , , . . . , n } , it is suffices to show the following conditions lim sup ǫ → ǫ log P ( ǫ,j ) x (cid:0) x ǫ,j ( τ j ) ∈ Γ j (cid:1) ≤ − I j (cid:0) x , ¯Γ j (cid:1) , (22)and lim inf ǫ → ǫ log P ( ǫ,j ) x (cid:0) x ǫ,j ( τ j ) ∈ Γ j (cid:1) ≥ − I j (cid:0) x , Γ ◦ j (cid:1) , (23)uniformly for x ∈ Ω and for any Γ j ⊂ ∂ Ω with ¯Γ j = ¯Γ ◦ j .Note that I j (cid:0) x , ¯Γ ◦ j (cid:1) = I j (cid:0) x , ¯Γ j (cid:1) (cf. (19)), then the upper bound in (22) can be verified usingthe Ventcel-Freidlin estimate (see [14, pp. 332–334] or [1]).On the other hand, to prove the lower bound in (23), we introduce a penalty function Φ M (with Φ M ( y ) = 0 for y ∈ Γ ); and write f ( j ) ( x ) = f ( j ) M ( x ) (cid:0) ≡ E ( ǫ,j ) x (cid:0) exp (cid:0) − ǫ Φ M (cid:0) x (cid:1)(cid:1)(cid:1)(cid:1) and J ( ǫ,j )Φ = J ( ǫ,j )Φ M ( x ) , with Φ j = Φ M . Then, from (17), we have q ( ǫ,j ) (cid:0) x , Γ j (cid:1) ≤ f ( j ) M ( x ) , (24)for each M . Hence, using Lemma 1 and noting further J ( ǫ,j )Φ M ( x ) ≥ I j ( x , Γ ◦ j ) , the lower boundin (23) holds uniformly for all x ∈ Λ . This completes the proof. ✷ In the following, using Proposition 1, we provide additional results on the exit positions of thestate-trajectories x ǫ,j ( t ) through the portion of the boundary Γ j for j = 0 , , . . . , n . For x, y ∈ ¯Ω , we consider the following I j ( ϕ, τ j ) = inf ϕ ∈H [0 ,T ] ,τ j ≥ Z τ j ∧ T L j (cid:0) ϕ ( t ) , ˙ ϕ ( t ) (cid:1) dt, ∀ j ∈ N ∪ { } , (25)where the infimum is taken among all ϕ ∈ H [0 , T ] and τ j ≥ such that ϕ (0) = x , ϕ ( τ j ) = y and ϕ ( t ) ∈ ¯Ω for all t ∈ [0 , τ j ∧ T ] . Then, using (17), we have I j (cid:0) x , Γ j (cid:1) = inf y ∈ Γ I j (cid:0) x , y (cid:1) , = min y ∈ ¯Γ I j (cid:0) x , y (cid:1) , j = 0 , , . . . , n, (26)for x ∈ Ω and Γ j ∈ ∂ Ω .Next, we will assume, in addition to Assumption 1, the followings (cf. [14, pp 359–360]). Assumption 2: (a)
D(cid:0) A + P ni =1 B i K i (cid:1) y, γ ( y ) E < and D(cid:0) A + P i = j B i K i (cid:1) y, γ ( y ) E < for j = 1 , , . . . , n ,where γ ( y ) is the unit outward normal to Ω at y ∈ ∂ Ω .(b) For all j ∈ { , , . . . , n } , let there exist a compact subset Λ ⊂ Ω such that:(i) I j ( x, y ) = 0 , ∀ x, y ∈ Λ .(ii) Let Λ δ denote the δ -neighborhood of Λ , and Ω δ = Ω \ ¯Λ δ . Then, there exists c δ thattends to zero as δ → such that I Ω δ j (cid:0) x, y (cid:1) ≤ I j (cid:0) x, y (cid:1) + c δ , ∀ x, y ∈ Ω \ Λ δ , (27)where the minimum functional I Ω δ j is with respect to Ω δ (cf. equation (7)).Notice that the statements in Assumption 2(b) imply the following I j (cid:0) x , y (cid:1) = I j (cid:0) x , y (cid:1) , ∀ x , x ∈ Λ , j = 0 , , . . . , n. (28)Hence, for each j = 0 , , . . . , n , if we let V j ( x, ∂ Ω) = inf y ∈ ∂ Ω I j (cid:0) x, y (cid:1) , x ∈ Λ , (29) and Σ j = n y ∈ ∂ Ω (cid:12)(cid:12)(cid:12) I j (cid:0) x, y (cid:1) = V j (cid:0) x, ∂ Ω (cid:1) , x ∈ Λ o . (30)Then, we immediately obtain the following proposition. Proposition 2:
Suppose that Assumptions 1 and 2 hold. Then, for any δ > , dist (cid:0) x ǫ,j , Σ j (cid:1) → in probability as ǫ → for each j = 0 , , . . . , n . Proof : For any fixed j ∈ { , , . . . , n } , let S be open, with smooth boundary, and Λ ⊂ S ⊂ Λ δ (where δ > is small enough such that ¯Λ δ ⊂ Ω ). Further, let Ω ¬ ¯ S = Ω \ ¯ S and let Γ j ⊂ ∂ Ω beclosed with Σ j ⊂ ¯Γ ◦ j and ¯Γ ◦ j = Γ j . Then, for any x ∈ Λ , there exits κ > such that I j (cid:0) x , Γ j (cid:1) = V j (cid:0) x , Σ j (cid:1) , (31)and I j (cid:0) x , Γ cj (cid:1) = V j (cid:0) x , Σ j (cid:1) + 2 κ, (32)where Γ cj = ∂ Ω \ Γ j for each j = 0 , , . . . , n .Note that, from Assumption 2(b), one can choose small δ > such that max z ∈ ∂ Λ δ I Ω ¬ ¯ S j (cid:0) z, Γ j (cid:1) < V j (cid:0) z, Σ j (cid:1) + κ < min z ∈ ∂ Λ δ I Ω ¬ ¯ S j (cid:0) z, Γ cj (cid:1) , (33)where the minimum functional I Ω ¬ ¯ S j is with respect to Ω ¬ ¯ S .Then, from Proposition 1, we have the following lim ǫ → q ( ǫ,j ) ¬ ¯ S (cid:0) z, Γ cj (cid:1) q ( ǫ,j ) ¬ ¯ S (cid:0) z, Γ j (cid:1) = 0 , j = 0 , , . . . , n, (34)uniformly for all z ∈ ∂ Λ δ .For x ∈ Ω (with x ǫ,j (0) = x ), let us define the following random time processes τ j, = inf n t (cid:12)(cid:12)(cid:12) x ǫ,j ( t ) ∈ Ω ¬ ¯ S o ,s j,k = inf n t (cid:12)(cid:12)(cid:12) t > τ j,k − , k ≥ , x ǫ,j ( t ) ∈ Ω ¬ ¯ S o ,τ j,k = inf n t (cid:12)(cid:12)(cid:12) t > s j,k , k ≥ , x ǫ,j ( t ) ∈ Ω ¬ ¯ S o . Next, we consider the following events A ( j ) k = n τ j = τ j,k , x ǫ,j ( τ j ) ∈ Γ j o , and B ( j ) k = n τ j = τ j,k , x ǫ,j ( τ j ) ∈ Γ cj o . Then, from the strong Markov property, we have P ( ǫ,j ) x ( A ( j ) k ) = E ( ǫ,j ) x (cid:16) χ τ j >s j,k q ( ǫ,j ) ¬ ¯ S (cid:0) x ǫ,j ( s j,k ) , Γ j (cid:1)(cid:17) , (35)and P ( ǫ,j ) x ( B ( j ) k ) = E ( ǫ,j ) x (cid:16) χ τ j >s j,k q ( ǫ,j ) ¬ ¯ S (cid:0) x ǫ,j ( s j,k ) , Γ cj (cid:1)(cid:17) , (36)where χ τ j >s j,k is an indicator function for the random event τ j > s j,k (with k ≥ ). Note that,from (35), for any ℓ > , there exits an ǫ ℓ > such that q ( ǫ,j ) ¬ ¯ S ( x , Γ cj ) ≤ ℓ q ( ǫ,j ) ¬ ¯ S ( x , Γ) , ∀ z ∈ ∂ Λ δ , ∀ ǫ ∈ (0 , ǫ ℓ ) . (37)Since x ǫ,j ( s j,k ) ∈ ∂ Λ δ , then we have P ( ǫ,j ) x (cid:16) B ( j ) k (cid:17) ≤ ℓ P ( ǫ,j ) x (cid:16) A ( j ) k (cid:17) . Moreover, we have X k P ( ǫ,j ) x (cid:16) A ( j ) k (cid:17) ≤ X k P ( ǫ,j ) x (cid:16) A ( j ) k [ B ( j ) k (cid:17) , = P ( ǫ,j ) x (cid:16) τ j < ∞ (cid:17)(cid:18) ≡ , for each j = 0 , , . . . , n (cid:19) . (38)Hence, for ǫ ∈ (0 , ǫ ℓ ) , we have (see also Footnote 6) P ( ǫ,j ) x (cid:16) x ǫ,j ( τ j ) ∈ Γ cj (cid:17) = X k P ( ǫ,j ) x (cid:16) B ( j ) k (cid:17) , ≤ P ( ǫ,j ) x (cid:16) B ( j )0 (cid:17) + ℓ, j = 0 , , . . . , n. (39)Since ℓ is arbitrary, this completes the proof. ✷ Note that the above proposition, i.e., Proposition 2, is connected to the boundary value problem,when one is also interested on the position of state-trajectories at the first time of their exit fromthe boundary Γ j ⊂ ∂ Ω . For the boundary value problem of Section II (cf. [14, pp. 371–272]),with L ( ǫ,j ) f ( j ) M ( x ) = 0 in Ω and f ( j ) M ( x ) = E ( ǫ,j ) x (cid:0) exp (cid:0) − ǫ Φ M (cid:0) x (cid:1)(cid:1)(cid:1) on ∂ Ω . For example, if Σ j Note that P ( ǫ,j ) x (cid:0) A ( j )0 S B ( j )0 (cid:1) → in probability as ǫ → for each j = 0 , , . . . , n . consists of a single point, say y ∗ , then f ( j ) M ( x ) → E ( ǫ,j ) x (cid:0) exp (cid:0) − ǫ Φ M (cid:0) y ∗ (cid:1)(cid:1)(cid:1) as ǫ → for all x ∈ Ω . Moreover, if Λ consists of a single point x ∗ ∈ Ω , then we have the following − lim ǫ → ǫ log E ( ǫ,j ) x ∗ (cid:16) x ǫ,j ( τ j ) ∈ Γ j (cid:17) = min y ∈ ∂ Ω I j ( x ∗ , y ) , = V j (cid:0) x ∗ , Σ j (cid:1) , j = 0 , , . . . , n, (40)which is equivalent to the result of Proposition 1. V. F
URTHER REMARKS
In this section, we briefly comment on the implication of our results on a co-design problem –when one is also interested in either evaluating the performance or finding a set of sub-optimalredundant controllers for the multi-channel system, while estimating the exit probabilities or theasymptotic bounds on the mean exit-time of the state-trajectories from the domain Ω . In particular, here we outline a multi-objective embedded optimization framework – where theproblem of optimal exit probabilities and finding a set of stabilizing feedbacks for the multi-channel system can be considered as a composite goal-oriented optimization problem (see, e.g.,[15]). Hence, the composite optimization problem (which also embeds additional subproblems)can be reformulated as follows min γ subject to I i (cid:0) x , Γ i (cid:1) − γ i w i ≤ I (cid:0) ϕ ∗ , τ ∗ , K ∗ (cid:1) , with Γ i ⊂ ∂ Ω & K ∗ ∈ K x ∈ Ω (initial condition) w i > , with P ni =1 w i = 1 γ i (unrestricted scalar variables) (41) Note that the last portion of the state-trajectories, prior reaching the boundary Γ j lies in the neighborhood of ϕ ( t ) ∈ H [0 , T ] for which I j ( x ∗ , Γ j ) differs little from V j (cid:0) x ∗ , Σ j (cid:1) for each j = 0 , , . . . , n . Note that, in Section IV, we provide estimates for the exit probabilities of state-trajectories from the boundary of the givendomain Ω for each particular operating condition (i.e., during the nominal operating condition or any single-channel failure inthe multi-channel system). where w i ’s are the weighting factors and the vector γ is given by [ γ , γ , . . . , γ n ] T . Moreover, I ( ϕ ∗ , τ ∗ , K ∗ ) , which corresponds to the nominal operating condition (i.e., without any fault inthe system), is given by I (cid:0) ϕ ∗ , τ ∗ , K ∗ (cid:1) = sup K ∈K inf ϕ ∈H [0 ,T ] ,τ ≥ Z τ L (cid:0) ϕ ( t ) , ˙ ϕ ( t ) (cid:1) dt, (42)where the I i (cid:0) x , Γ i (cid:1) ’s (together with the boundary conditions x ǫ,i ( τ i ) ∈ Γ i for i = 1 , , . . . , n )are assumed to satisfy the optimization subproblems in (26) (cf. equation (40)). Note that suchclass of stabilizing state-feedbacks can further be restricted to satisfy additional assumptions.Here, we remark that the max-min problem in (42) for the exit probabilities has been studiedin the past (see, e.g., [8] in the context of differential games for a general admissible class ofcontrols; and see also [6] or [7] via viscosity solution techniques). Remark 3:
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