Optimal control problems with state constraints for semilinear distributed-parameter systems

We consider optimal control problems for distributed-parameter systems described by semilinear equations, with constraints on the control and on the state, and an exact pointwise target condition. As an application of a general theory of nonlinear programming problems in Banach spaces, a version of the Pontryagin maximum principle is obtained.This research was partly supported by the National Science Foundation under Grant DMS-92-21819.     

Optimal control of a singular PDE modeling transient MEMS with control or state constraints

A particular feature of certain microelectromechanical systems (MEMS) is the appearance of a so-called “pull-in” instability, corresponding to a singularity in the underlying PDE model. We here consider a transient MEMS model and its optimal control via the dielectric properties of the membrane and/or the applied voltage. In contrast to the static case, the control problem suffers from low dimensionality of the control compared to the state and hence requires different techniques for establishing first order optimality conditions. For this purpose, we here use a relaxation approach combined with a localization technique.     

Optimal feedback control for a linear-quadratic control problem with a state inequality constraint

In this paper, we consider the linear-quadratic control problem (LQCP) for systems defined by evolution operators with an inequality constraint on the state. It is shown that, under suitable assumptions, the optimal control exists, is unique, and has a closedloop structure. The synthesis of the feedback control requires one to solve two Riccati integral equations.     

Necessary extremum conditions in the optimal control problem with state constraints

The optimal control problem with state constraints is examined. An alternative to the available approaches to the study of this problem is proposed. The maximum principle and second-order necessary conditions are proved.     

Optimal boundary control of hyperbolic equations with pointwise state and mixed control‐state constraints

This paper is devoted to deal with optimal control problems of semilinear hyperbolic equations with pointwise state constraints and mixed control‐state constraints. We obtain necessary optimality conditions in the form of a Pontryagin's minimum principle. Our approach is based on modern methods of variational analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

Penalty approximation of a Sobolev optimal control problem with state constraints

In this paper, we study the approximation by the penalty method of a control problem governed by a pseudo-parabolic equation with a noncoercive control functional and with control and state constraints. The existence of solutions to the penalized problems is established. In addition, the convergence of the penalized problems to the solution, the Lagrange multipliers, and the minimum value of the original problem is studied. The results apply to Sobolev and parabolic equations as well.This work was partially supported by the National Science Foundation, Grant No. MCS-79-02037. The author would like to thank Professor A. B. Schwarzkopf for his helpful comments on this paper.     

On optimal control problems with state constraints

This paper is concerned with necessary conditions for a general optimal control problem developed by Russak and Tan. It is shown that, in most cases, a further relation between the multipliers holds. This result is of interest in particular for the investigation of perturbations of the state constraint.     

A solution method for regular optimal control problems with state constraints

A method of region analysis is developed for solving a class of optimal control problems with one state and one control variable, including state and control constraints. The performance index is strictly convex with respect to the control variable, while this variable appears only linearly in the state equation. The convexity or linearity assumption of the performance index or the state equation with respect to the state variable is not required.The author would like to express his sincere gratitude to Prof. R. Klötzler, Prof. E. Zeidler, Prof. H. Schumann, Prof. J. Focke, and other colleagues of the Department of Mathematics, Karl Marx University, Leipzig, GDR, for their support during his stay in Leipzig.     

A maximum principle for smooth optimal impulsive control problems with multipoint state constraints

A nonlinear optimal impulsive control problem with trajectories of bounded variation subject to intermediate state constraints at a finite number on nonfixed instants of time is considered. Features of this problem are discussed from the viewpoint of the extension of the classical optimal control problem with the corresponding state constraints. A necessary optimality condition is formulated in the form of a smooth maximum principle; thorough comments are given, a short proof is presented, and examples are discussed.     

Sufficient optimality conditions for nonconvex control problems with state constraints

The sufficient optimality conditions of Zeidan for optimal control problems (Refs. 1 and 2) are generalized such that they are applicable to problems with pure state-variable inequality constraints. We derive conditions which neither assume the concavity of the Hamiltonian nor the quasiconcavity of the constraints. Global as well as local optimality conditions are presented.     

Some results on optimal control with unilateral state constraints

In this paper, we study the problem of quadratic optimal control with state variables unilateral constraints, for linear time-invariant systems. The necessary conditions are formulated as a linear invariant system with complementary slackness conditions. Some structural properties of this system are examined. Then it is shown that the problem can benefit from the higher order Moreau’s sweeping process, that is, a specific distributional differential inclusion, and from ten Dam’s geometric theory [A.A. ten Dam, K.F. Dwarshuis, J.C. Willems, The contact problem for linear continuous-time dynamical systems: A geometric approach, IEEE Trans. Automat. Control 42 (4) (1997) 458–472; A.A. ten Dam, Unilaterally Constrained Dynamical Systems, Ph.D. Thesis, Rijsuniversiteit Groningen, NL, available at http://irs.ub.rug.nl/ppn/159407869, 1997] for partitioning of the admissible domain boundary (in particular for the case of multivariable systems). In fact, the first step may be also seen as follows: does the higher order Moreau’s sweeping process (developed in Acary et al. [V. Acary, B. Brogliato, D. Goeleven, Higher order Moreau’s sweeping process: Mathematical formulation and numerical simulation, Math. Programm. A 113 (2008) 133–217]) correspond to the necessary conditions of some optimal control problem with an extended integral action? The knowledge of the qualitative behaviour of optimal trajectories at junction times is improved with the approach, which also paves the way towards efficient time-stepping numerical algorithms to solve the optimal control boundary value problem.     

Optimal solutions to differential inclusions in presence of state constraints

Necessary conditions in terms of the Hamiltonian are given for optimal solutions to the differential inclusion problem when state constraints are present. This result extends a result of Clarke for the unconstrained problem. The data are nonsmooth, nonlinear, nonconvex. The method incorporates the state constraint in the cost functional as a penalty term for a sequence of unconstrained problems that approximate our problem. An application of Ekeland's variational principle, the known necessary conditions for the auxiliary problems, and a limiting process provide the necessary conditions.     

求解带界约束的非线性方程组的混合方法

基于非单调技术和L-M算法, 提出了一种新的求解带界约束的非线性方程组的混合方法. 在一定条件下, 该算法具有全局收敛性. 数值试验表明该算法是有效的.     

Optimal control in nonlinear infinite-dimensional systems with nondifferentiability of two types

We consider the optimal control problem for systems described by nonlinear equations of elliptic type. If the nonlinear term in the equation is smooth and the nonlinearity increases at a comparatively low rate of growth, then necessary conditions for optimality can be obtained by well-known methods. For small values of the nonlinearity exponent in the smooth case, we propose to approximate the state operator by a certain differentiable operator. We show that the solution of the approximate problem obtained by standard methods ensures that the optimality criterion for the initial problem is close to its minimal value. For sufficiently large values of the nonlinearity exponent, the dependence of the state function on the control is nondifferentiable even under smoothness conditions for the operator. But this dependence becomes differentiable in a certain extended sense, which is sufficient for obtaining necessary conditions for optimality. Finally, if there is no smoothness and no restrictions are imposed on the nonlinearity exponent of the equation, then a smooth approximation of the state operator is possible. Next, we obtain necessary conditions for optimality of the approximate problem using the notion of extended differentiability of the solution of the equation approximated with respect to the control, and then we show that the optimal control of the approximated extremum problem minimizes the original functional with arbitrary accuracy.     

A transformation technique for optimal control problems with partially linear state inequality constraints

This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector.A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector.In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-76-3075, and by the National Science Foundation, Grant No. MCS-76-21657.     

The Lagrange-Newton method for state constrained optimal control problems

Local convergence of the Lagrange-Newton method for optimization problems with two-norm discrepancy in abstract Banach spaces is investigated. Based on stability analysis of optimization problems with two-norm discrepancy, sufficient conditions for local superlinear convergence are derived. The abstract results are applied to optimal control problems for nonlinear ordinary differential equations subject to control and state constraints.This research was completed while the second author was a visitor at the University of Bayreuth, Germany, supported by grant No. CIPA3510CT920789 from the Commission of the European Communities.     

The Lagrange-Newton method for nonlinear optimal control problems

We investigate local convergence of the Lagrange-Newton method for nonlinear optimal control problems subject to control constraints including the situation where the terminal state is fixed. Sufficient conditions for local quadratic convergence of the method based on stability results for the solutions of nonlinear control problems are discussed.     

Asymptotic method for solving the time-optimal control problem for a nonlinear singularly perturbed system

The time-optimal control problem for a nonlinear singularly perturbed system with multidimensional controls bounded in the Euclidean norm is considered. An algorithm for constructing asymptotic approximations to its solution is proposed. The main advantage of the algorithm is that the original optimal control problem decomposes into two unperturbed problems of lower dimensions.     

Optimal control of parabolic problems with state constraints: a penalization method for optimality conditions

In this paper state constrained optimal control problems governed by parabolic evolution equations are studied. Our purpose is to obtain a (first-order) decoupled optimality system (that ensures the Lagrange multipliers existence). In a first step we are led to Slater-like assumptions and we are then allowed to extend the application field of the decoupled system we obtain. With a weaker assumption the existence of Lagrange multipliers (that are measures) for nonqualified problems may be established.