Existence and properties of solutions of a control system with hysteresis effect

We consider a control system described by two ordinary nonlinear differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values, which depends on the phase variables. One of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase variable. The equation containing the subdifferential describes an input-output relation of hysteresis type.Along with the original control constraint, we also consider the convexified control constraint and the constraint consisting of the extremal points of the convexified control constraint.We prove the existence of solutions of our control system with various control constraints and establish certain relationships between corresponding solution sets.     

Properties of Solutions to Second Order Evolution Control Systems. I

We consider a control system described by a nonlinear second order evolution equation defined on an evolution triple of Banach spaces (Gelfand triple) with a mixed multivalued control constraint whose values are nonconvex closed sets. Alongside the original system we consider a system with the following control constraints: a constraint whose values are the closed convex hull of the values of the original constraint and a constraint whose values are extreme points of the constraint which belong simultaneously to the original constraint. By a solution to the system we mean an admissible trajectory-control pair. In this part of the article we study existence questions for solutions to the control system with various constraints and density of the solution set with nonconvex constraints in the solution set with convexified constraints.     

Properties of the set of “trajectory-control” pairs of a control systemwith subdifferential operators

We consider a control system described by an evolution equation with control constraint which is a multivalued mapping of a phase variable with closed nonconvex values. One of the evolution operators of the system is the subdifferential of a time-dependent proper, convex, and lower semicontinuous function. The other operator, acting on the derivative of the required functions, is the subdifferential of a convex continuous function. We also consider systems with the following control constraints: multivalued mappings whose values are the closed convex hulls of the values of the original constraint and multivalued mapping whose values are the extreme points of the convexified constraint that belong to the original one. We study topological properties of the sets of admissible “trajectory–control” pairs of the system with various control constraints and clarify the relations between them. An example of a parabolic system with hysteresis and diffusion phenomena is considered in detail. Bibliography: 19 titles.     

Solutions of evolution inclusions generated by a difference of subdifferentials

An evolution inclusion with the right-hand side containing the difference of subdifferentials of proper convex lower semicontinuous functions and a multivalued perturbation whose values are nonconvex closed sets is considered in a separable Hilbert space. In addition to the original inclusion, we consider an inclusion with convexified perturbation and a perturbation whose values are extremal points of the convexified perturbation that also belong to the values of the original perturbation. Questions of the existence of solutions under various perturbations are studied and relations between solutions are established. The primary focus is on the weakening of assumptions on the perturbation as compared to the known assumptions under which existence and relaxation theorems are valid. All our assumptions, in contrast to the known assumptions, concern the convexified rather than original perturbation.     

Existence and relaxation of solutions to differential inclusions with unbounded right-hand side in a Banach space

In a separable Banach space we consider a differential inclusion whose values are nonconvex, closed, but not necessarily bounded sets. Along with the original inclusion, we consider the inclusion with convexified right-hand side. We prove existence theorems and establish relations between solutions to the original and convexified differential inclusions. In contrast to assuming that the right-hand side of the inclusion is Lipschitz with respect to the phase variable in the Hausdorff metric, which is traditional in studying this type of questions, we use the (ρ–H) Lipschitz property. Some example is given.     

Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system

The problem of minimization of an integral functional with an integrand that is nonconvex with respect to the control is considered. We minimize our functional over the solution set of a nonlinear evolution control system with a time-dependent subdifferential operator in a Hilbert space. The control constraint is given by a nonconvex closed bounded set. The integrand, the control constraint, the initial conditions and the operators in the equation describing the control system all depend on a parameter. We consider, along with the original problem, the problem of minimizing an integral functional with an integrand convexified with respect to the control over the solution set of the same system, but now subject to the convexified control constraint. By a solution of the control system we mean a “trajectory–control” pair. We prove that for each value of the parameter the convexified problem has a solution, which is the limit of a minimizing sequence of the original problem, and the minimum value of the functional of the convexified problem is a continuous function of the parameter.     

Minimal trajectories of nonconvex differential inclusions

We consider an optimization problem with endpoint constraints associated with a nonconvex differential inclusion. We give a necessary condition of the maximum principle type for a solution of the problem. Following the approach from Ref. 1, the condition is stated in terms of single-valued selections of the convexified right-hand side of the inclusion.This work was supported in part by the National Science Foundation, Grant No. DMS-86-01774.     

On minima of a functional of the gradient: Upper and lower solutions

This paper studies a scalar minimization problem with an integral functional of the gradient under affine boundary conditions. A new approach is proposed using a minimal and a maximal solution to the convexified problem. We prove a density result: any relaxed solution continuously depending on boundary data may be approximated uniformly by solutions of the nonconvex problem keeping continuity relative to data. We also consider solutions to the nonconvex problem having Lipschitz dependence on boundary data with the best Lipschitz constant.     

Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations

In this paper, we study optimal relaxed controls and relaxation of nonlinear fractional impulsive evolution equations. Firstly, existence of piecewise continuous mild solutions for the original fractional impulsive control system is presented. Secondly, fractional impulsive relaxed control system is constructed by using a regular countably additive measure and making the original control system convexified. Thirdly, optimal relaxed controls and relaxation theorems are obtained. Finally, application to initial-boundary value problem of fractional impulsive parabolic control system is considered.     

Relaxation Results for Hybrid Inclusions

The Filippov–Wa?ewski relaxation theorem describes when the set of solutions to a differential inclusion is dense in the set of solutions to the relaxed (convexified) differential inclusion. This paper establishes relaxation results for a broad range of hybrid systems which combine differential inclusions, difference inclusions, and constraints on the continuous and discrete motions induced by these inclusions. The relaxation results are used to deduce continuous dependence on initial conditions of the sets of solutions to hybrid systems.     

Subdifferential inclusions with unbounded perturbation: Existence and relaxation theorems

The paper studies an evolution inclusion in a separable Hilbert space whose right-hand side contains the subdifferential of a proper convex lower semicontinuous function of time and a set-valued perturbation. Together with this inclusion, an inclusion with convexified perturbation values is considered. The existence and density of the solution set of the initial inclusion in the closure of the solution set of the inclusion with convexified perturbation are proved. This property is usually called relaxation. Traditional assumptions for relaxation theorems are the compactness property of the convex function and the boundedness of the perturbation. In the present paper, such assumptions are not made. Assumptions for subdifferential inclusions described by polyhedral sweeping processes and variational inequalities with time-dependent obstacles and constraints are specified.     

Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints: Part 1. Boundary Control

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve these problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.     

Properties of the Set of Extreme Solutions for a Class of Nonlinear Second-Order Evolution Inclusions

We establish the existence of extreme solutions for a class of nonlinear second-order evolution inclusions with a nonconvex right-hand side defined on an evolution triple of Banach spaces. Then we show that extreme solutions which belong to the solution set of the original system are in fact dense and codense in the solution set of a system with a convexified right-hand side. The necessary and sufficient conditions for closedness of the solution set for the original system in an appropriate spaces of functions are given as well. Finally, an example of a nonlinear hyperbolic distributed parameter system is worked out in detail.     

On the “bang-bang” principle for nonlinear evolution inclusions

We establish the existence of extreme solutions for a class of nonlinear evolution inclusions with non-convex right-hand side defined on an evolution triple of Banach spaces. Then we show that extreme solutions which belong to the solution set of the original system are in fact dense in the solutions of the system with convexified right-hand side. Subsequently we use this density result to derive nonlinear and infinite-dimensional version of the “bang-bang” principle for control systems. An example of a nonlinear parabolic distributed parameter system is also worked out in detail. Received November 21, 1997     

Properties of Solutions to Second Order Evolution Control Systems. II

We continue the research of the first part of the article. We mainly study codensity for the set of admissible trajectory-control pairs of a system with nonconvex constraints in the set of admissible trajectory-control pairs of the system with convexified constraints. We state necessary and sufficient conditions for the set of admissible trajectory-control pairs of a system with nonconvex constraints to be closed in the corresponding function spaces. Using an example of a control hyperbolic system, we give an interpretation of the abstract results obtained. As application we consider the minimization problem for an integral functional on solutions of a control system.     

Relaxation in the optimal control problem for the Goursat-Darboux system

We consider the optimal control problem for a system described by the Goursat-Darboux equation. The system is controlled by distributed and boundary controls satisfying mixed nonconvex constraints. For this problem we prove an analog of the classical Bogolyubov relaxation theorem.     

Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations

We introduce the optimality question to the relaxation in multiple control problems described by Sobolev-type nonlinear fractional differential equations with nonlocal control conditions in Banach spaces. Moreover, we consider the minimization problem of multi-integral functionals, with integrands that are not convex in the controls, of control systems with mixed nonconvex constraints on the controls. We prove, under appropriate conditions, that the relaxation problem admits optimal solutions. Furthermore, we show that those optimal solutions are in fact limits of minimizing sequences of systems with respect to the trajectory, multicontrols, and the functional in suitable topologies.     

Asymptotic optimization of nonlinear singularly perturbed systems with control constrained by a hypersphere

We consider a terminal control problem for a nonlinear singularly perturbed system with constraints imposed on the right endpoint of the trajectories. The values of the multidimensional controls are bounded in the Euclidean norm. We suggest an algorithm for an approximate (in the asymptotic sense) solution of this problem. The key advantage of the algorithm is the following: the original problem splits into two optimal control problems of lower dimension, one of which is linear.     

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.     

Alternating boundary layer type solutions of some singularly perturbed periodic parabolic equations with Dirichlet and Robin boundary conditions

In this paper, we continue the analysis of alternating boundary layer type solutions to certain singularly perturbed parabolic equations for which the degenerate equations (obtained by setting small parameter multiplying derivatives equal to zero) are algebraic equations that have three roots. Here, we consider spatially one-dimensional equations. We address special cases where the following are true: (a) boundary conditions are of the Dirichlet type with different values of unknown functions specified at different endpoints of the interval of interest; (b) boundary conditions are of the Robin type with an appropriate power of a small parameter multiplying the derivative in the conditions. We emphasize a number of new features of alternating boundary layer type solutions that appear in these cases. One of the important applications of such equations is related to modeling certain types of bioswitches. Special choices of Dirichlet and Robin type boundary conditions can be used to tune up such bioswitches. This article was submitted by the authors in English.