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Full-Text Articles in Physical Sciences and Mathematics
Optimal Control Of Delay-Differential Inclusions With Multivalued Initial Conditions In Infinite Dimensions, Boris S. Mordukhovich, Dong Wang, Lianwen Wang
Optimal Control Of Delay-Differential Inclusions With Multivalued Initial Conditions In Infinite Dimensions, Boris S. Mordukhovich, Dong Wang, Lianwen Wang
Mathematics Research Reports
This paper is devoted to the study of a general class of optimal control problems described by delay-differential inclusions with infinite-dimensional state spaces, endpoints constraints, and multivalued initial conditions. To the best of our knowledge, problems of this type have not been considered in the literature, except some particular cases when either the state space is finite-dimensional or there is no delay in the dynamics. We develop the method of discrete approximations to derive necessary optimality conditions in the extended Euler-Lagrange form by using advanced tools of variational analysis and generalized differentiation in infinite dimensions. This method consists of the …
Discrete Approximations, Relaxation, And Optimization Of One-Sided Lipschitzian Differential Inclusions In Hilbert Spaces, Tzanko Donchev, Elza Farkhi, Boris S. Mordukhovich
Discrete Approximations, Relaxation, And Optimization Of One-Sided Lipschitzian Differential Inclusions In Hilbert Spaces, Tzanko Donchev, Elza Farkhi, Boris S. Mordukhovich
Mathematics Research Reports
We study discrete approximations of nonconvex differential inclusions in Hilbert spaces and dynamic optimization/optimal control problems involving such differential inclusions and their discrete approximations. The underlying feature of the problems under consideration is a modi- fied one-sided Lipschitz condition imposed on the right-hand side (i.e., on the velocity sets) of the differential inclusion, which is a significant improvement of the conventional Lipschitz continuity. Our main attention is paid to establishing efficient conditions that ensure the strong approximation (in the W^1,p-norm as p greater than or equal to 1) of feasible trajectories for the one-sided Lipschitzian differential inclusions under. consideration by …
Discrete Approximations Of Differential Inclusions In Infinite-Dimensional Spaces, Boris S. Mordukhovich
Discrete Approximations Of Differential Inclusions In Infinite-Dimensional Spaces, Boris S. Mordukhovich
Mathematics Research Reports
In this paper we study discrete approximations of continuous-time evolution systems governed by differential inclusions with nonconvex compact values in infinite-dimensional spaces. Our crucial result ensures the possibility of a strong Sobolev space approximation of every feasible solution to the continuous-time inclusion by its discrete-time counterparts extended as Euler's "broken lines." This result allows us to establish the value and strong solution convergences of discrete approximations of the Bolza problem for constrained infinite-dimensional differential/evolution inclusions under natural assumptions on the initial data.
Optimal Control Of Semilinear Evolution Inclusions Via Discrete Approximations, Boris S. Mordukhovich, Dong Wang
Optimal Control Of Semilinear Evolution Inclusions Via Discrete Approximations, Boris S. Mordukhovich, Dong Wang
Mathematics Research Reports
This paper studies a Mayer type optimal control problem with general endpoint constraints for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces. First, we construct a sequence of discrete approximations to the original optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions …
Optimal Control Of Delay Systems With Differential And Algebraic Dynamic Constraints, Boris S. Mordukhovich, Lianwen Wang
Optimal Control Of Delay Systems With Differential And Algebraic Dynamic Constraints, Boris S. Mordukhovich, Lianwen Wang
Mathematics Research Reports
This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential inclusion and a linear constraint link between "slow" and "fast" variables. We pursue a two-hold goal: to study variational stability for this class of control systems with respect to discrete …
Optimal Control Of Delayed Differential-Algebraic Inclusions, Boris S. Mordukhovich, Lianwen Wang
Optimal Control Of Delayed Differential-Algebraic Inclusions, Boris S. Mordukhovich, Lianwen Wang
Mathematics Research Reports
This paper concerns constrained dynamic optimization problems governed by delayed differential-algebraic systems. Dynamic constraints in such systems, which are particularly important for engineering applications, are described by interconnected delay-differential inclusions and algebraic equations. We pursue a two-hold goal: to study variational stability of such control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. We are not familiar with any results in these directions for differential-algebraic inclusions even in the delay-free case. In the first part of the paper …
The Approximate Maxium Principle In Constrained Optimal Control, Boris S. Mordukhovich, Ilya Shvartsman
The Approximate Maxium Principle In Constrained Optimal Control, Boris S. Mordukhovich, Ilya Shvartsman
Mathematics Research Reports
The paper concerns optimal control problems for dynamic systems governed by a parametric family of discrete approximations of control systems with continuous time. Discrete approximations play an important role in both qualitative and numerical aspects of optimal control and occupy an intermediate position between discrete-time and continuous-time control systems. The central result in optimal control of discrete approximations is the Approximate Maximum Principle (AMP), which is justified for smooth control problems with endpoint constraints under certain assumptions without imposing any convexity, in contrast to discrete systems with a fixed step. We show that these assumptions are essential for the validity …
Neumann Boundary Control Of Hyperbolic Equations With Pointwise State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond
Neumann Boundary Control Of Hyperbolic Equations With Pointwise State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond
Mathematics Research Reports
We consider optimal control problems for hyperbolic systems with controls in Neumann boundary conditions with pointwise (hard) constraints on control and state functions. Focusing on hyperbolic dynamics governed by the multidimensional wave equation with a nonlinear term, we derive new necessary optimality conditions in the pointwise form of the Pontryagin Maximum Principle for the state-constrained problem under consideration. Our approach is based on modern methods of variational analysis that allows us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional.
Dirichlet Boundary Control Of Hyperbolic Equations In The Presence Of State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond
Dirichlet Boundary Control Of Hyperbolic Equations In The Presence Of State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond
Mathematics Research Reports
We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.
Optimal Control Of Neutral Functional-Differential Inclusions, Boris S. Mordukhovich, Lianwen Wang
Optimal Control Of Neutral Functional-Differential Inclusions, Boris S. Mordukhovich, Lianwen Wang
Mathematics Research Reports
This paper deals with optimal control problems for dynamical systems governed by constrained functional-differential inclusions of neutral type. Such control systems contain time-delays not only in state variables but also in velocity variables, which make them essentially more complicated than delay-differential (or differential-difference) inclusions. Our main goal is to derive necessary optimality conditions for general optimal control problems governed by neutral functional-differential inclusions with endpoint constraints. While some results are available for smooth control systems governed by neutral functional-differential equations, we are not familiar with any results for neutral functional-differential inclusions, even with smooth cost functionals in the absence of …
Discrete Maximum Principle For Nonsmooth Optimal Control Problems With Delays, Boris S. Mordukhovich, Ilya Shvartsman
Discrete Maximum Principle For Nonsmooth Optimal Control Problems With Delays, Boris S. Mordukhovich, Ilya Shvartsman
Mathematics Research Reports
We consider optimal control problems for discrete-time systems with delays. The main goal is to derive necessary optimality conditions of the discrete maximum principle type in the case of nonsmooth minimizing functions. We obtain two independent forms of the discrete maximum principle with transversality conditions described in terms of subdifferentials and superdifferentials, respectively. The superdifferential form is new even for non-delayed systems and may be essentially stronger than a more conventional subdifferential form in some situations.