Approximations of Relaxed Optimal Control Problems

In the present paper, we investigate an approximation technique for relaxed optimal control problems. We study control processes governed by ordinary differential equations in the presence of state, target, and integral constraints. A variety of approximation schemes have been recognized as powerful tools for the theoretical studying and practical solving of Infinite-dimensional optimization problems. On the other hand, theoretical approaches to the relaxed optimal control problem with constraints are not sufficiently advanced to yield numerically tractable schemes. The explicit approximation of the compact control set makes it possible to reduce the sophisticated relaxed problem to an auxiliary optimization problem. A given trajectory of the relaxed problem can be approximated by trajectories of the auxiliary problem. An optimal solution of the introduced optimization problem provides a basis for the construction of minimizing sequences for the original optimal control problem. We describe how to carry out the numerical calculations in the context of nonlinear programming and establish the convergence properties of the obtained approximations.The authors thank the referees for helpful comments and suggestions.     

Some remarks on the numerical solution of bang-bang type optimal control problems

This paper is concerned with the numerical solution of optimal control problems for which each optimal control is bang-bang. Especially, the results apply to parabolic boundary control Problems. Starting from a sequence of feasible solutions converging to an optimal control u, a sequence of bang-bang controls converging to u is constructed. Bang-bang approximations of u are desirable for certain numerical reasons. Sequences of arbitrary feasible controls converging to u may be obtained by discretization or by a descent method. Numerical examples are also given.     

Existence of Optimal Controls in the Absence of Cesari-Type Conditions forSemilinear Elliptic and Parabolic Systems

Some new results on the existence of optimal controls are established for control systems governed by semilinear elliptic or parabolic equations. No Cesari type conditions are assumed. By proving existence theorems and analyzing the Pontryagin maximum principle for optimal relaxed state-control pairs for the corresponding relaxed problems, existence theorems of classical optimal pairs for the original problem are established. To treat the case of a noncompact control set, relaxed controls defined by finitely additive measures are used.     

Approximation of relaxed nonlinear parabolic optimal control problems

We consider a relaxed optimal control problem for systems defined by nonlinear parabolic partial differential equations with distributed control. The problem is completely discretized by using a finite-element approximation scheme with piecewise linear states and piecewise constant controls. Existence of optimal controls and necessary conditions for optimality are derived for both the continuous and the discrete problem. We then prove that accumulation points of sequences of discrete optimal [resp. extremal] controls are optimal [resp. extremal] for the continuous problem.     

Discrete approximation of relaxed optimal control problems

We consider a general nonlinear optimal control problem for systems governed by ordinary differential equations with terminal state constraints. No convexity assumptions are made. The problem, in its so-called relaxed form, is discretized and necessary conditions for discrete relaxed optimality are derived. We then prove that discrete optimality [resp., extremality] in the limit carries over to continuous optimality [resp., extremality]. Finally, we prove that limits of sequences of Gamkrelidze discrete relaxed controls can be approximated by classical controls.     

Relaxation methods for mixed-integer optimal control of partial differential equations

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments.     

Mixed Frank–Wolfe Penalty Method with Applications to Nonconvex Optimal Control Problems

We consider a general optimization problem which is an abstract formulation of a broad class of state-constrained optimal control problems in relaxed form. We describe a generalized mixed Frank–Wolfe penalty method for solving the problem and prove that, under appropriate assumptions, accumulation points of sequences constructed by this method satisfy the necessary conditions for optimality. The method is then applied to relaxed optimal control problems involving lumped as well as distributed parameter systems. Numerical examples are given.     

Necessary conditions for optimal control problems with bounded state by a penalty method

Necessary conditions are derived for a general relaxed control problem with unilateral state constraint. The results are also valid for ordinary controls that are solutions of the relaxed problem.A penalty is imposed to change the constrained problem into a sequence of unconstrained problems. The assumptions are on the data of the problem and do not requirea priori verification of hypotheses involving the optimal solution.     

Optimal Design of the Time-Dependent Support of Bang–Bang Type Controls for the Approximate Controllability of the Heat Equation

We consider the nonlinear optimal shape design problem, which consists in minimizing the amplitude of bang–bang type controls for the approximate controllability of a linear heat equation with a bounded potential. The design variable is the time-dependent support of the control. Precisely, we look for the best space–time shape and location of the support of the control among those, which have the same Lebesgue measure. Since the admissibility set for the problem is not convex, we first obtain a well-posed relaxation of the original problem and then use it to derive a descent method for the numerical resolution of the problem. Numerical experiments in 2D suggest that, even for a regular initial datum, a true relaxation phenomenon occurs in this context. Also, we implement a simple algorithm for computing a quasi-optimal domain for the original problem from the optimal solution of its associated relaxed one.     

Convergent computational method for relaxed optimal control problems

A class of relaxed optimal control problems for ordinary differential equations with a state-space constraint is considered. The discretization by the control parametrization method, formerly proposed by Teo and Goh (Refs. 1, 2), is modified by admitting a tolerance in the state constraint, which enables one to prove a conditional convergence under certain additional qualification on the dynamics. Also, a counterexample is constructed, showing that the original, nonmodified discretization need not approximate the continuous problem.The author is grateful to Professor K. L. Teo for useful comments on this paper.     

Bounded state problem for hereditary control problems

In this paper, we characterize optimal pairs for a hereditary control problem where the state is constrained. We use relaxed controls and the technique of penalization.This research was supported by NSF Grant No. HRD-91-54077.     

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.     

Numerical Simulation of the Two-Hydrodynamic Film Thickness

We investigate, from a numerical point of view, the coefficients identification problem, for the Elrod–Adams model of cavitation, in the frame of the hydrodynamic lubrication. We relax the control problem, and propose a relaxed augmented Lagrangian algorithm, with a given loop length. Some numerical results are given.     

Nonexistence and Existence of an Optimal Control Problem Governed by a Class of Semilinear Elliptic Equations

An optimal control problem governed by a class of semilinear elliptic equations is considered in this paper. Using relaxed controls, the nonexistence and existence results of an optimal control are obtained.     

Optimal Control Problems of Phase Relaxation Models

This work is concerned with optimal control problems with convex cost criterion governed by the relaxed Stefan problem with or without memory. The existence of an optimal control is proved and necessary conditions for a given function to be an optimal control are found. Moreover, an asymptotic analysis is performed as the time relaxation parameter tends to zero.     

Global optimization of polynomial-expressed nonlinear optimal control problems with semidefinite programming relaxation

In this paper, we propose a new deterministic global optimization method for solving nonlinear optimal control problems in which the constraint conditions of differential equations and the performance index are expressed as polynomials of the state and control functions. The nonlinear optimal control problem is transformed into a relaxed optimal control problem with linear constraint conditions of differential equations, a linear performance index, and a matrix inequality condition with semidefinite programming relaxation. In the process of introducing the relaxed optimal control problem, we discuss the duality theory of optimal control problems, polynomial expression of the approximated value function, and sum-of-squares representation of a non-negative polynomial. By solving the relaxed optimal control problem, we can obtain the approximated global optimal solutions of the control and state functions based on the degree of relaxation. Finally, the proposed global optimization method is explained, and its efficacy is proved using an example of its application.     

Controllability,extremality, and abnormality in nonsmooth optimal control

We derive sufficient conditions for controllability and necessary conditions for minimum in nonsmooth optimal control problems defined by differential or functional-integral equations with isoperimetric and unilateral restrictions. We consider the cases when the controls are relaxed or chosen fromabundant sets of original (ordinary) controls (which include most, or all, of the control sets studied in the literature). We prove that, if there exist optimal strictly original controls (that is, controls that are optimal in an abundant set but not among relaxed controls), then the problem admits abnormal extremals. We also study the abnormality of the optimal strictly original controls themselves.     

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

We consider a controlled system driven by a coupled forward–backward stochastic differential equation with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation, at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition.     

Coupled Galerkin and parameterization methods for optimal control of discretely connected parallel beams

Galerkin and wavelet methods for optimal boundary control of a couple of discretely connected parallel beams are proposed. First, the problem with boundary controls is converted into a problem with distributed controls. The problem is, then, reduced by a Galerkin-based approach into determining the optimal control of a linear time-invariant lumped parameter system, which will be solved by a wavelet-based method using Legendre wavelets. The integration-operational matrix and Kronecker product are utilized to significantly simplify the optimization problem into a system of linear equations. A numerical example is presented to demonstrate the applicability and the efficiency of the proposed method.     

Numerical solution of a relaxed control problem by the epsilon method

A relaxed control problem is solved using the epsilon method of Balakrishnan. The singularity of the problem with respect to one of the controls presents no difficulty. However, solutions are sensitive to the initial guess of the states and controls. For a system with true dynamics, terminal conditions may generally be approximated, but cannot be satisfied exactly.This paper is based on the MS Thesis of K. Shmueli at the Department of Aeronautical Engineering, Technion-Israel Institute of Technology, Haifa, Israel.