Unbounded Differential Inclusions Research Articles

Extended Euler–Lagrange and Hamiltonian Conditions in Optimal Control of Sweeping Processes with Controlled Moving Sets

This paper concerns optimal control problems for a class of sweeping processes governed by discontinuous unbounded differential inclusions that are described via normal cone mappings to controlled moving sets. Largely motivated by applications to hysteresis, we consider a general setting where moving sets are given as inverse images of closed subsets of finite-dimensional spaces under nonlinear differentiable mappings dependent on both state and control variables. Developing the method of discrete approximations and employing generalized differential tools of first-order and second-order variational analysis allow us to derive nondegenerate necessary optimality conditions for such problems in extended Euler–Lagrange and Hamiltonian forms involving the Hamiltonian maximization. The latter conditions of the Pontryagin Maximum Principle type are the first in the literature for optimal control of sweeping processes with control-dependent moving sets.

Necessary Optimality Conditions for the Mayer Problem with Unbounded Differential Inclusion

Abstract Based on the properties of solutions of a differential inclusion with unbounded measurable-pseudo-Lipschitz right-hand side, the necessary conditions in the Mayer extremal problem on set of trajectories of the unbounded differential inclusion are obtained. DOI: 10.1134/S0081543813080099

Optimality conditions for a controlled sweeping process with applications to the crowd motion model

The paper concerns the study and applications of a new class of optimal control problems governed by a perturbed sweeping process of the hysteresis type with control functions acting in both play-and-stop operator and additive perturbations. Such control problems can be reduced to optimization of discontinuous and unbounded differential inclusions with pointwise state constraints, which are immensely challenging in control theory and prevent employing conventional variation techniques to derive necessary optimality conditions. We develop the method of discrete approximations married with appropriate generalized differential tools of modern variational analysis to overcome principal difficulties in passing to the limit from optimality conditions for finite-difference systems. This approach leads us to nondegenerate necessary conditions for local minimizers of the controlled sweeping process expressed entirely via the problem data. Besides illustrative examples, we apply the obtained results to an optimal control problem associated with of the crowd motion model of traffic flow in a corridor, which is formulated in this paper. The derived optimality conditions allow us to develop an effective procedure to solve this problem in a general setting and completely calculate optimal solutions in particular situations.

Optimal control of a perturbed sweeping process via discrete approximations

The paper addresses an optimal control problem for a perturbed sweeping process of the rate-independent hysteresis type described by a controlled ``play-and stop operator with separately controlled perturbations. This problem can be reduced to dynamic optimization of a state-constrained unbounded differential inclusion with highly irregular data that cannot be treated by means of known results in optimal control theory for differential inclusions. We develop the method of discrete approximations, which allows us to adequately replace the original optimal control problem by a sequence of well-posed finite-dimensional optimization problems whose optimal solutions strongly converge to that of the controlled perturbed sweeping process. To solve the discretized control systems, we derive effective necessary optimality conditions by using second-order generalized differential tools of variational analysis that explicitly calculated in terms of the given problem data.

Optimal control of the sweeping process over polyhedral controlled sets

The paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given Bolza-type functional, which depends on control and state variables as well as their velocities. Besides the highly non-Lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process, the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types. All of this creates serious challenges for deriving necessary optimality conditions. We develop here the method of discrete approximations and combine it with advanced tools of first-order and second-order variational analysis and generalized differentiation. This approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions. As a by-product of the developed approach, we prove the strong W1,2-convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous-time system and derive necessary optimality conditions for the discrete counterparts. The established necessary optimality conditions for the sweeping process are illustrated by several examples.

Time Optimum Problems for Unbounded Differential Inclusion

Based on the properties of solutions of a differential inclusion with unbounded measurable-pseudo-Lipschitz right-hand side, the necessary conditions in time optimum problem on set of solutions of the unbounded differential inclusion are obtained. DOI: 10.1134/S0081543813080099

Stratified Necessary Conditions for Differential Inclusions with State Constraints

The concept of stratified necessary conditions for optimal control problems, whose dynamic constraint is formulated as a differential inclusion, was introduced by F. H. Clarke. These are conditions satisfied by a feasible state trajectory that achieves the minimum value of the cost over state trajectories whose velocities lie in a time-varying open ball of specified radius about the velocity of the state trajectory of interest. Considering different radius functions stratifies the interpretation of “minimizer.” In this paper we prove stratified necessary conditions for optimal control problems involving pathwise state constraints. As was shown by Clarke in the state constraint-free case, we find that, also in our more general setting, the stratified necessary conditions yield generalizations of earlier optimality conditions for unbounded differential inclusions as simple corollaries. Some examples are provided, giving insights into the nature of the hypotheses invoked for the derivation of stratified necessary conditions and into the scope for their further refinement.

Optimal control of semilinear unbounded differential inclusions

This paper concerns a general optimal control problem of the Mayer type for semilinear unbounded differential inclusions with endpoint constraints in reflexive and separable Banach spaces. Based on the method of discrete approximations and advanced generalized differential tools of variational analysis, we establish necessary optimality conditions for discrete-time control problems and then, by passing to the limit, derive intrinsic optimality conditions of the extended Euler–Lagrange type for semilinear evolution inclusions.

Sufficient Conditions for Error Bounds and Applications

Our aim in this paper is to present sufficient conditions for error bounds in terms of Frechet and limiting Frechet subdifferentials in general Banach spaces. This allows us to develop sufficient conditions in terms of the approximate subdifferential for systems of the form (x, y) ∈ C × D, g(x, y, u) = 0, where g takes values in an infinite-dimensional space and u plays the role of a parameter. This symmetric structure offers us the choice of imposing conditions either on C or D. We use these results to prove the nonemptiness and weak-star compactness of Fritz–John and Karush–Kuhn–Tucker multiplier sets, to establish the Lipschitz continuity of the value function and to compute its subdifferential and finally to obtain results on local controllability in control problems of nonconvex unbounded differential inclusions.

Cosmically Lipschitz set-valued mappings

This paper introduces a kind of sub-Lipschitz continuity for set-valued mappings based on the cosmic metric. This type of Lipschitz behavior has applications with regards to necessary optimality conditions, the Hamilton–Jacobi equation, and invariance of unbounded differential inclusions. Cosmically Lipschitz assumptions allow for broader applications than previously allowed under Lipschitz assumptions. It is also shown that a cosmically Lipschitz mapping can be characterized by the normal cones to its graph using the coderivative, and various rules are presented in order to more easily identify such a mapping.

Discontinuous Solutions to Unbounded Differential Inclusions under State Constraints. Applications to Optimal Control Problems

We consider a nonconvex and unbounded differential inclusion derived from a control system whose control sets are time and space-dependent. We extend the inclusion in order to allow discontinuous trajectories. We prove that the set of solutions of the original inclusion is dense in the set of solutions of the extended inclusion and, moreover, these last solutions are stable with respect to the initial data. Both of these results are also proven in the presence of state and integral constraints (assuming suitable conditions at the boundary of the constraining set). As an application, the value function of a Mayer problem is shown to be continuous and the unique viscosity solution of a Hamilton–Jacobi equation with suitable boundary conditions.

Exit time problems for nonlinear unbounded control systems

Given a control system (formulated as a nonconvex and unbounded differential inclusion) we study the problem of reaching a closed target with trajectories of the system. A controllability condition around the target allows us to construct a path that steers each point nearby into it in finite time and using a finite amount of energy. In applications to minimization problems, limits of such trajectories could be discontinuous. We extend the inclusion so that all the trajectories of the extension can be approached by (graphs of) solutions of the original system. In the extended setting the value function of an exit time problem with Lagrangian affine in the unbounded control can be shown to coincide with the value function of the original problem, to be continuous and to be the unique (viscosity) solution of a Hamilton-Jacobi equation with suitable boundary conditions.

Second-order necessary conditions for differential inclusion problems

We study second-order necessary conditions for optimality in the unbounded differential inclusion control problem and recover the accessory problem in optimal control theory.

Optimal Control of Unbounded Differential Inclusions

A Mayer problem of optimal control, whose dynamic constraint is given by a convex-valued differential inclusion, is considered. Both state and endpoint constraints are involved. Necessary conditions are proved incorporating the Hamiltonian inclusion, the Euler–Lagrange inclusion, and the Weierstrass–Pontryagin maximum condition. These results weaken the hypotheses and strengthen the conclusions of earlier works. Their main focus is to allow the admissible velocity sets to be unbounded, provided they satisfy a certain continuity hypothesis. They also sharpen the assertion of the Euler–Lagrange inclusion by replacing Clarke’s subgradient of the essential Lagrangian with a subset formed by partial convexification of limiting subgradients. In cases where the velocity sets are compact, the traditional Lipschitz condition implies the continuity hypothesis mentioned above, the assumption of “integrable boundedness” is shown to be superfluous, and this refinement of the Euler–Lagrange inclusion remains a strict improvement on previous forms of this condition.