On a class of nonsmooth optimal control problems

In the paper the Zangwill-Pietrzykowski penalties are used for the augmentation of state-space and terminal state inequality constraints in an optimal control problem with a nonsmooth cost of a special structure having a straightforward practical application. Directional derivatives of the augmented cost are derived. In such a way bundle methods for nonsmooth minimization may be utilized.     

On a class of inverse optimal control problems

Four distinct, though closely related, inverse optimal control problems are considered. Given a closed, convex setU in a real Hilbert spaceX and an elementu ₀ inU, it is desired to find all functionals of the form (u,Ru) such that (i)R is a self-adjoint positive operator and (u,Ru) is minimized over the setU at the pointu ₀, (ii)R is self-adjoint, positive definite and (u,Ru) is minimized overU atu ₀, (iv)R is self-adjoint, positive definite and (u,Ru) is uniquely minimized overU atu ₀. The interrelationships among the sets of solutions of these problems are pointed out. Necessary and sufficient conditions which explicitly characterize the solutions to each of these problems are derived. The question of existence of a solution (namely, Given a particular setU and a particular elementu ₀, under what conditions does there exist an operatorR having certain required properties?) is discussed. The results derived are illustrated by an example.     

A class of optimal control problems for piezoelectric frictional contact models

We consider control problems for a mathematical model describing the frictional bilateral contact between a piezoelectric body and a foundation. The material’s behavior is modeled with a linear electro–elastic constitutive law, the process is static and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity conditions on the contact surface are described with the Clarke subdifferential boundary conditions. The weak formulation of the problem consists of a system of two hemivariational inequalities. We provide the results on existence and uniqueness of a weak solution to the model and, under additional assumptions, the continuous dependence of a solution on the data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.     

A computational method for a class of optimal relaxed control problems

In the present paper, we propose a computational scheme for solving a class of optimal relaxed control problems, using the concept of control parametrization. Furthermore, some important convergence properties of the proposed computational scheme are investigated. For illustration, a numerical example is also included.     

A control reduced primal interior point method for a class of control constrained optimal control problems

A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparatively coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples. Supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin. This paper appeared as ZIB Report 04-38.     

A stochastic optimal control approach to a class of production and inventory problems

A comprehensible and unified system control approach is presented to solve a class of production/inventory smoothing problems. A nonstationary, non-Gaussian, finite-time linear optimal solution with an attractive computation scheme is obtained for a general quadratic and linear cost structure. A complete solution to a classical production/inventory control problem is given as an example. A general solution to the discrete-time optimal regulator with arbitrary but known disturbance is provided and discussed in detail. A computationally attractive closed-loop suboptimal scheme is presented for problems with constraints or nonquadratic costs. Implementation and interpretation of the results are discussed.     

On a class of optimal control problems for parabolic equations

Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 100–1005, September–October, 1994.     

On a class of nonconvex and nonlinear optimal control problems

We propose two relaxation approaches for the existence of solutions of a nonconvex optimal control problem with a nonlinear dynamics and two fixed endpoints. The first approach adds to the functional a term depending on the final state; the second one introduces a new scalar control into both the functional and the dynamics. Our results generalize those obtained in the linear case. We assume that the integrand be concave w.r.t. the state variable, and provide an example showing that a strict concavity condition, in the nonlinear case, is essential. Finally, a result without relaxation is presented.     

Analysis and semidiscrete galerkin approximation of A class of nonlinear hyperbolic optimal control problems

This paper presents an optimization problem involving a system governed by a nonlinear hyperbolic equation with control acting on the domain, in the boundary conditions, or in the initial conditions. The quadratic cost function is the classical one. We present the Galerkin approximation and we demonstrate existence of the weak and strong condensation points of a set of solutions of the approximate optimization problems. Each of these points is a solution of the initial optimization problem     

Function space approach to a class of stochastic optimal control problems

The solution of the linear time-dependent stochastic optimal control problem with a quadratic performance index is well known. This paper presents a method of using this known solution to solve a class of problems which have the same system equations but with a class of non-quadratic performance indices.This work was partially supported by the Jet Propulsion Laboratory, California Institute of Technology, NASA Contract No. NAS 7-100.     

A class of nondifferentiable control problems

Optimality conditions and duality results are obtained for a class of control problems having a nondifferentiable term in the integrand of the objective functional. These results generalize many well-known results in optimal control theory involving differentiable functions, and also provide a relationship with certain nondifferentiable mathematical programming problems. Some extensions concerning the unified treatment of optimal control theory and continuous programming are also mentioned. Finally, a control problem containing an arbitrary norm, along with its appropriate norm, is given.     

Euler discretization for a class of nonlinear optimal control problems with control appearing linearly

We investigate Euler discretization for a class of optimal control problems with a nonlinear cost functional of Mayer type, a nonlinear system equation with control appearing linearly and constraints defined by lower and upper bounds for the controls. Under the assumption that the cost functional satisfies a growth condition we prove for the discrete solutions Hölder type error estimates w.r.t. the mesh size of the discretization. If a stronger second-order optimality condition is satisfied the order of convergence can be improved. Numerical experiments confirm the theoretical findings.     

On a class of optimal control problems with distributed and lumped parameters

The optimal control of moving sources governed by a parabolic equation and a system of ordinary differential equations with initial and boundary conditions is considered. For this problem, an existence and uniqueness theorem is proved, sufficient conditions for the Fréchet differentiability of the cost functional are established, an expression for its gradient is derived, and necessary optimality conditions in the form of pointwise and integral maximum principles are obtained.     

Approximation of a class of optimal control problems with order of convergence estimates

An approximation scheme for a class of optimal control problems is presented. An order of convergence estimate is then developed for the error in the approximation of both the optimal control and the solution of the control equation.     

A computational method for a class of non-standard time optimal control problems involving multiple time horizons

In this paper, we consider a class of non-standard time optimal control problems involving a dynamical system consisting of multiple subsystems evolving over different time horizons. Different subsystems are required to reach their respective target sets at different termination times. The goal is to minimize the maximum of these termination times. By introducing a discrete variable to represent the system termination ordering, we reformulate this problem as a discrete optimization problem. A discrete filled function method is developed to solve this discrete optimization problem. For illustration, a numerical example is solved.     

Existence of solutions for a class of nonconcave infinite horizon optimal control problems

In this article, we establish the existence of optimal solutions for a large class of nonconvex infinite horizon discrete-time optimal control problems. This class contains optimal control problems arising in economic dynamics which describe a model with nonconcave utility functions representing the preferences of the planner.     

Numerical algorithm for a class of constrained optimal control problems of switched systems

In this paper, we consider an optimal control problem of switched systems with continuous-time inequality constraints. Because of the complexity of such constraints and switching laws, it is difficult to solve this problem by standard optimization techniques. To overcome the difficulty, we adopt a bi-level algorithm to divide the problem into two nonlinear constrained optimization problems: one continuous and the other discrete. To solve the problem, we transform the inequality constraints into equality constraints which is smoothed using a twice continuously differentiable function and treated as a penalty function. On this basis, the smoothed problem can be solved by any second-order gradient algorithm, e.g., Newton’s Method. Finally, numerical examples show that our method is effective compared to existing algorithms.     

A virtual control concept for state constrained optimal control problems

A linear elliptic control problem with pointwise state constraints is considered. These constraints are given in the domain. In contrast to this, the control acts only at the boundary. We propose a general concept using virtual control in this paper. The virtual control is introduced in objective, state equation, and constraints. Moreover, additional control constraints for the virtual control are investigated. An error estimate for the regularization error is derived as main result of the paper. The theory is illustrated by numerical tests.