Vera Zeidan

First- and second-order necessary conditions for control problems with constraints

Second-order necessary conditions are developed for an abstract nonsmooth control problem with mixed state-control equality and inequality constraints as well as a constraint of the form G(x, u) E IF, where F is a closed convex set of a Banach space with nonempty interior. The inequality constraints g(s, x, u) < 0 depend on a parameter s belonging to a compact metric space S. The equality constraints are split into two sets of equations K(x, u) = 0 and H(x, u) = 0, where the first equation is an abstract control equation, and H is assumed to have a full rank property in u. The objective function is maxtET f(t, x, u) where T is a compact metric space, f is upper semicontinuous in t and Lipschitz in (x, u). The results are in terms of a function a that disappears when the parameter spaces T and S are discrete. We apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints. Thus we obtain a generalization and extension of the existing results on this problem.

Symplectic difference systems: variable stepsize discretization and discrete quadratic functionals

Discrete quadratic functionals with variable endpoints for variable stepsize symplectic difference systems are considered. A comprehensive study is presented for characterizing the positivity of such functionals in terms of conjugate intervals, conjoined bases, and implicit and explicit Riccati equations with various forms of boundary conditions. Moreover, necessary conditions for the nonnegativity of these functionals are obtained in terms of the above notions. Furthermore, we show that a variable stepsize discretization of a continuous-time nonlinear control problem leads to a discrete linear quadratic problem and a Hamiltonian difference system, which are special cases of their symplectic counterparts.

Optimal Control Problems with Set-Valued Control and State Constraints

In this paper a general optimal control problem with pure state and mixed control-state constraints is considered. These constraints are of the form of set-inclusions. Second-order necessary optimality conditions for weak local minimum are derived for this problem in terms of the original data. In particular the nonemptiness of the set of critical directions and the evaluation of its support function are expressed in terms of the given functions and set-valued maps. In order that the Lagrange multiplier corresponding to the mixed control-state inclusion constraint be represented via an integrable function, a strong normality condition involving the notion of the critical tangent cone is introduced.

Coupled points in the calculus of variations and applications to periodic problems

The aim of this paper is to introduce the definition of coupled points for the problems of the calculus of variations with general boundary conditions, and to develop second order necessary conditions for optimality. When one of the end points is fixed, our necessary conditions reduce to the known ones involving conjugate points. We also apply our results to the periodic problems of the calculus of variations.

Second Order Sufficiency Criteria for a Discrete Optimal Control Problem

In this work, we derive second order necessary and sufficient optimality conditions for a discrete optimal control problem with one variable endpoint and the other fixed, and with equality control constraints. In particular, the positivity of the second variation, which is a discrete quadratic functional with appropriate boundary conditions, is characterized in terms of the nonexistence of intervals conjugate to 0, the existence of a certain conjoined basis of the associated linear Hamiltonian difference system, or the existence of a symmetric solution to the implicit and explicit Riccati matrix equations. Some results require a certain minimal normality assumption, and are derived using the sensitivity analysis technique.

Coupled points in optimal control theory

The concept of coupled points is introduced for an optimal control problem where both state endpoints are allowed to vary. This definition leads to the extension of the theory of conjugate points to the optimal-control setting. Under suitable controllability assumptions, weaker than those previously considered, it is shown that the nonexistence of coupled points in the open interval (a, b) is a necessary condition for weak local optimality. This result generalizes the ones of the same kind known from the calculus of variations. In the special case when one or both state endpoints are fixed, the notion of coupled points is more general than those of focal or conjugate points. >

Time scale embedding theorem and coercivity of quadratic functionals

In this paper we study the relation between the coercivity and positivity of a time scale quadratic functional J, which could be a second variation for a nonlinear time scale calculus of variations problem (P). We prove for the case of general jointly varying endpoints that J is coercive if and only if it is positive definite and the time scale version of the strengthened Legendre condition holds. In order to prove this, we establish a time scale embedding theorem and apply it to the Riccati matrix equation associated with the quadratic functional J. Consequently, we obtain sufficiency criteria for the nonlinear problem (P) in terms of the positivity of J or in terms of the time scale Riccati equation. This result is new even for the continuous time case when the endpoints are jointly varying.

Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: survey

In this paper we provide a survey of characterizations of the nonnegativity and positivity of discrete quadratic functionals which arise as the second variation for nonlinear discrete calculus of variations problems. These characterizations are in terms of (i) (strict) conjugate and (strict) coupled intervals, (ii) the conjoined bases of the associated Jacobi difference equation, and (iii) the solution of the corresponding Riccati difference equation. The results depend on the form of the boundary conditions of the quadratic functional and, basically, we distinguish three types: (a) separable endpoints with zero right endpoint (this of course includes the simplest case of both zero endpoints), (b) separable endpoints, and (c) jointly varying endpoints.

Discrete optimal control: Second order optimality conditions

In this paper, we present a survey and refinement of our recent results in the discrete optimal control theory. For a general nonlinear discrete optimal control problem (P) , second order necessary and sufficient optimality conditions are derived via the nonnegativity ( I S 0) and positivity ( I >0) of the discrete quadratic functional I corresponding to its second variation. Thus, we fill the gap in the discrete-time theory by connecting the discrete control problems with the theory of conjugate intervals, Hamiltonian systems, and Riccati equations. Necessary conditions for I S 0 are formulated in terms of the positivity of certain partial discrete quadratic functionals, the nonexistence of conjugate intervals, the existence of conjoined bases of the associated linear Hamiltonian system, and the existence of solutions to Riccati matrix equations. Natural strengthening of each of these conditions yields a characterization of the positivity of I and hence, sufficiency criteria for the original problem (P) . Finally, open problems and perspectives are also discussed.

Coupled intervals in the discrete calculus of variations:necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.