Sufficient Optimality Conditions for Optimal Control Problems with State Constraints

It is well-known in optimal control theory that the maximum principle, in general, furnishes only necessary optimality conditions for an admissible process to be an optimal one. It is also well-known that if a process satisfies the maximum principle in a problem with convex data, the maximum principle turns to be likewise a sufficient condition. Here an invexity type condition for state constrained optimal control problems is defined and shown to be a sufficient optimality condition. Further, it is demonstrated that all optimal control problems where all extremal processes are optimal necessarily obey this invexity condition. Thus optimal control problems which satisfy such a condition constitute the most general class of problems where the maximum principle becomes automatically a set of sufficient optimality conditions.     

On Global Optimality Conditions for Nonlinear Optimal Control Problems

Let a trajectory and control pair maximize globally the functional g(x(T)) in the basic optimal control problem. Then (evidently) any pair (x,u) from the level set of the functional g corresponding to the value g( (T)) is also globally optimal and satisfies the Pontryagin maximum principle. It is shown that this necessary condition for global optimality of turns out to be a sufficient one under the additional assumption of nondegeneracy of the maximum principle for every pair (x,u) from the above-mentioned level set. In particular, if the pair satisfies the Pontryagin maximum principle which is nondegenerate in the sense that for the Hamiltonian H, we have along the pair on [0,T], and if there is no another pair (x,u) such that g(x(T))=g( (T)), then is a global maximizer.     

Optimal Bang-Bang Controls for a Two-Compartment Model in Cancer Chemotherapy

A class of mathematical models for cancer chemotherapy which have been described in the literature take the form of an optimal control problem over a finite horizon with control constraints and dynamics given by a bilinear system. In this paper, we analyze a two-dimensional model in which the cell cycle is broken into two compartments. The cytostatic agent used as control to kill the cancer cells is active only in the second compartment where cell division occurs and the cumulative effect of the drug is used to model the negative effect of the treatment on healthy cells. It is shown that singular controls are not optimal for this model and the optimality properties of bang-bang controls are established. Specifically, transversality conditions at the switching surfaces are derived. In a nondegenerate setting, these conditions guarantee the local optimality of the flow if satisfied, while trajectories will be no longer optimal if they are violated.     

Second-Order Sufficient Conditions for State-Constrained Optimal Control Problems

Second-order sufficient optimality conditions (SSC) are derived for an optimal control problem subject to mixed control-state and pure state constraints of order one. The proof is based on a Hamilton-Jacobi inequality and it exploits the regularity of the control function as well as the associated Lagrange multipliers. The obtained SSC involve the strict Legendre-Clebsch condition and the solvability of an auxiliary Riccati equation. They are weakened by taking into account the strongly active constraints.     

Necessary Optimality Conditions for a Dead Oil Isotherm Optimal Control Problem

We study a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of the mechanics of a continuous medium. Recent results on the problem provide existence, uniqueness and regularity of the optimal solution. Here we obtain the first necessary optimality conditions. Work supported by the Portuguese Foundation for Science and Technology (FCT) through the Centre for Research in Optimization and Control (CEOC) of the University of Aveiro, cofinanced by the European Community fund FEDER/POCTI. The first author was also supported by the postdoctoral fellowship SFRH/BPD/20934/2004.     

Sufficient Optimality Conditions in Stability Analysis for State-Constrained Optimal Control

A family of parametric linear-quadratic optimal control problems is considered. The problems are subject to state constraints. It is shown that if weak second-order sufficient optimality conditions and standard constraint qualifications are satisfied at the reference point, then, for small perturbations of the parameter, there exists a locally unique stationary point, corresponding to a solution. This point is a Lipschitz continuous function of the parameter.     

Verification of Second-Order Sufficient Optimality Conditions for Semilinear Elliptic and Parabolic Control Problems

We study optimal control problems for semilinear parabolic equations subject to control constraints and for semilinear elliptic equations subject to control and state constraints. We quote known second-order sufficient optimality conditions (SSC) from the literature. Both problem classes, the parabolic one with boundary control and the elliptic one with boundary or distributed control, are discretized by a finite difference method. The discrete SSC are stated and numerically verified in all cases providing an indication of optimality where only necessary conditions had been studied before.     

Sufficient conditions for a control to be a strong minimum

The control literature either presents sufficient conditions for global optimality (for example, the Hamilton-Jacobi-Bellman theorem) or, if concerned with local optimality, restricts attention to comparison controls which are local in theL -sense. In this paper, use is made of an exact expression for the change in cost due to a change in control, a natural extension of a result due to Weierstrass, to obtain sufficient conditions for a control to be a strong minimum (in the sense that comparison controls are merely required to be close in theL ₁-sense).     

Weak and Strong Optimality Conditions for Constrained Control Problems with Discontinuous Control

In recent years, sufficient optimality criteria and solution stability in optimal control have been investigated widely and used in the analysis of discrete numerical methods. These results were concerned mainly with weak local optima, whereas strong optimality has been considered often as a purely theoretical aspect. In this paper, we show via an example problem how weak the weak local optimality can be and derive new strong optimality conditions. The criteria are suitable for practical verification and can be applied to the case of discontinuous controls with changes in the set of active constraints.     

A Note on First-Order Sufficient Optimality Conditions for Pareto Problems

We take into consideration the first-order sufficient conditions, established by Jiménez and Novo (Numer. Funct. Anal. Optim. 2002; 23:303–322) for strict local Pareto minima. We give here a more operative condition for a strict local Pareto minimum of order 1.     

Sufficiency for Optimal Control Problems Involving Parameters

The optimal control problem is extended to the case where the performance index, the differential constraints, and the prescribed final conditions contain parameters. The sufficient condition for a minimum is derived for nonsingular problems using the sweep method. As expected, it involves the finiteness of a matrix or the location of the conjugate point. The minimum-time navigation problem is solved as a fixed final time problem to illustrate the application of the theory.     

Optimality Conditions for Nonregular Optimal Control Problems and Duality

We define a new class of optimal control problems and show that this class is the largest one of control problems where every admissible process that satisfies the Extended Pontryaguin Maximum Principle is an optimal solution of nonregular optimal control problems. In this class of problems the local and global minimum coincide. A dual problem is also proposed, which may be seen as a generalization of the Mond–Weir-type dual problem, and it is shown that the 2-invexity notion is a necessary and su?cient condition to establish weak, strong, and converse duality results between a nonregular optimal control problem and its dual problem. We also present an example to illustrate our results.     

Fast Solvers of Fredholm Optimal Control Problems

<正>The formulation of optimal control problems governed by Fredholm integral equations of second kind and an efficient computational framework for solving these control problems is presented.Existence and uniqueness of optimal solutions is proved. A collective Gauss-Seidel scheme and a multigrid scheme are discussed.Optimal computational performance of these iterative schemes is proved by local Fourier analysis and demonstrated by results of numerical experiments.     

Sufficient conditions for asymptotic optimal control

For a selected family of Lagrange-type control problems involving a nonnegative integral costJ _T (y,u) over the interval [0,T], 0<T<, with system conditions consisting of differential inequalities and/or equalities, the following material is treated: (i) a resumé of relevant necessary conditions and sufficient conditions for a pair (y _T ,u _T ) to minimizeJ _T (y,u); (ii) conditions sufficient for the convergence asT of minimizing pairs (y _T ,u _T ) over [0,T] to a limit pair (y ,u ) over the infinite-time interval [0, ); (iii) conditions sufficient for (y ,u ) to minimize the costJ (y,u) over [0, ); and (iv) conditions sufficient for the optimal cost per unit timeJ _T (y _T ,u _T )/T to have a limit asT.     

Optimal Control of Differential–Algebraic Equations of Higher Index,Part 2: Necessary Optimality Conditions

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of problems which can be transformed to index one control problems. For these problems we show in the accompanying paper that, if the solutions to the adjoint equations are well–defined, then the first-order approximations to the functionals defining the problem can be expressed in terms of the adjoint variables. In this paper we show that the solutions to the adjoint equations are essentially bounded measurable functions. Then, based on the first order approximations, we derive the necessary optimality conditions for the considered class of control problems. These conditions do not require the transformation of the DAEs to index-one system; however, higher-index DAEs and their associated adjoint equations have to be solved.     

Necessary Optimality Conditions for Optimal Control Problems with Nonsmooth Mixed State and Control Constraints

In this paper we study an optimal control problem with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality condition for optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition and under the assumption that the state and control constraint functions are smooth. In this paper we derive necessary optimality conditions for problems with nonsmooth mixed state and control constraints under constraint qualifications based on pseudo-Lipschitz continuity and calmness of certain set-valued maps. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. Moreover necessary optimality conditions with an Euler inclusion taking an explicit multiplier form are derived for certain cases.     

Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without a priori Normality Assumptions

First-order and second-order necessary conditions of optimality for an impulsive control problem that remain informative for abnormal control processes are presented and derived. One of the main features of these conditions is that no a priori normality assumptions are required. This feature follows from the fact that these conditions rely on an extremal principle which is proved for an abstract minimization problem with equality constraints, inequality constraints, and constraints given by an inclusion in a convex cone. Two simple examples illustrate the power of the main result.The first author was partially supported by the Russian Foundation for Basic Research Grant 02-01-00334. The second author was partially supported by the Russian Foundation for Basic Research Grant 00-01-00869. The third author was partially supported by Fundacao para a Ciencia e Tecnologia and by INVOTAN Grant.     

Error Estimates for a Class of Elliptic Optimal Control Problems

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.     

Optimality Conditions for State-Constrained Dirichlet Boundary Control Problems

In this paper, we prove that the combination of representation theorems for additive measures introduced in Ref. 1, together with a Lagrange multiplier theorem, leads to a short and direct proof of the optimality conditions for Dirichlet control problems with pointwise state constraints.