Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.     

On the unimprovability of full-memory strategies in problems of guaranteed result optimization

A problem of guaranteed result optimization is considered for a control system described by an ordinary differential equation and for a performance functional that depends continuously on the trajectory of the system. The values of the control and of the disturbance satisfy compact geometric constraints. It is also assumed that the realization of the disturbance is subject to an unknown functional constraint from a given set of constraints that are compact in a Lebesgue space. It is shown that the optimal guaranteed result in the class of full-memory strategies in this problem coincides with the value of the optimal guaranteed result in the class of quasi-strategies.     

State-Constrained Optimal Control Problems of Impulsive Differential Equations

The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption.     

Nonlinear optimal control problems of degenerate parabolic equations with logistic time-varying delays of convolution type

In this article, we consider a bioeconomic model for optimal control problems which are governed by degenerate parabolic equations governing diffusive biological species with logistic growth terms and multiple time-varying delays. The time-varying delays are given in a convolution form. The existence, uniqueness and regularity results to the state equations with homogeneous Dirichlet and Neumann boundary conditions are established. The vanishing viscosity method is used to obtain the existence result. Afterwards, we formulate the optimal control problem in two cases. Firstly, we suppose that this biological species causes damage to environment (e.g. forest, agriculture): the optimal control is the trapping rate and the cost functional is a combination of damage and trapping costs. Secondly, an optimal harvesting control of a biological species is considered: the optimal control is a distribution of harvesting effort on the biological species and the cost functional measure the difference between economic revenue and cost. The existence and the condition of uniqueness of the optimal solution are obtained. A nonlinear optimality system is derived, characterizing the optimal control.     

Regularity of solutions for an optimal control problem with mixed control-state constraints

A class of optimal control problems for a semilinear elliptic partial differential equation with mixed control-state constraints is considered. Existence results of an optimal control and necessary optimality conditions are stated. Moreover, a projection formula is derived that is equivalent to the necessary optimality conditions. As main result, the Lipschitz continuity of the optimal control is obtained.     

The Necessary Conditions of Fractional Optimal Control in the Sense of Caputo

This paper deals with the optimal control problem of a fractional dynamic system in the sense of Caputo. The main result of this paper gives a second order necessary optimality condition for fractional optimal control problems, which has not been discussed before. An application is introduced to explain our main results.     

Optimal Control Problem Governed by Semilinear Parabolic Equation and its Algorithm

In this paper, an optimal control problem governed by semilinear parabolic equation which involves the control variable acting on forcing term and coefficients appearing in the higher order derivative terms is formulated and analyzed. The strong variation method, due originally to Mayne et al to solve the optimal control problem of a lumped parameter system, is extended to solve an optimal control problem governed by semilinear parabolic equation, a necessary condition is obtained, the strong variation algorithm for this optimal control problem is presented, and the corresponding convergence result of the algorithm is verified.     

Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem

We study the numerical approximation of distributed nonlinear optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Our main result are error estimates for optimal controls in the maximum norm. Characterization results are stated for optimal and discretized optimal control. Moreover, the uniform convergence of discretized controls to optimal controls is proven under natural assumptions.     

Control with a Guide in the Guarantee Optimization Problem under Functional Constraints on the Disturbance

A motion control problem for a dynamic system under disturbances is considered on a finite time interval. There are compact geometric constraints on the values of the control and disturbance. The equilibrium condition in the small game is not assumed. The aim of the control is to minimize a given terminal performance index. The guaranteed result optimization problem is posed in the context of the game-theoretical approach. In the case when realizations of the disturbance belong to some a priori unknown compact subset of L₁ (the space of functions that are Lebesgue summable with the norm), we propose a new discrete-time control procedure with a guide. The proximity between the motions of the system and the guide is provided by the dynamic reconstruction of the disturbance. The quality of the control process is achieved by using an optimal counter-strategy in the guide. Conditions on the equations of motion under which this procedure ensures an optimal guaranteed result in the class of quasi-strategies are given. The scheme of the proof makes it possible to estimate the deviation of the realized value of the performance index from the value of the optimal result depending on the discretization parameter. Illustrative examples are given.     

Convergence of a feasible directions algorithm for a distributed optimal control problem of parabolic type with terminal inequality constraints

In this paper, we consider a class of optimal control problems with control and terminal inequality constraints, where the system dynamics is governed by a linear second-order parabolic partial differential equation with first boundary condition. A feasible direction algorithm for solving this class of optimal control problems has already been obtained in the literature. The aim of this paper is to improve the convergence result by using a topology arising in the study of relaxed controls.     

Duality for Set-Valued Multiobjective Optimization Problems,Part 1: Mathematical Programming

The duality of multiobjective problems is studied with the help of the apparatus of conjugate set-valued mappings introduced by the author. In this paper (Part 1), a duality theory is developed for set-valued mappings, which is then used to derive dual relations for some general multiobjective optimization problems which include convex programming and optimal control problems. Using this result, in the companion paper (Part 2), duality theorems are proved for multiobjective quasilinear and linear optimal control problems. The theory is applied to get dual relations for some multiobjective optimal control problem.     

Linear square optimal control problem for stochastic difference equations with unknown parameters

The problems of stability and optimal control for stochastic difference equations are receiving important attention now (see, for example, [1–3]). In this paper, the optimal control in final form is obtained for optimal control problem of stochastic linear difference equation with unknown parameters and square cost functional. For stochastic functional differential equations, analogous result are obtained in [4].     

Some Facts about the Ramsey Model

In modeling the dynamics of capital, the Ramsey equation coupled with the Cobb–Douglas production function is reduced to a linear differential equation by means of the Bernoulli substitution. This equation is used in the optimal growth problem with logarithmic preferences. The study deals with solving the corresponding infinite horizon optimal control problem. We consider a vector field of the Hamiltonian system in the Pontryagin maximum principle, taking into account control constraints. We prove the existence of two alternative steady states, depending on the constraints. This result enriches our understanding of the model analysis in the optimal control framework.     

On an Augmented Lagrangian SQP Method for a Class of Optimal Control Problems in Banach Spaces

An augmented Lagrangian SQP method is discussed for a class of nonlinear optimal control problems in Banach spaces with constraints on the control. The convergence of the method is investigated by its equivalence with the generalized Newton method for the optimality system of the augmented optimal control problem. The method is shown to be quadratically convergent, if the optimality system of the standard non-augmented SQP method is strongly regular in the sense of Robinson. This result is applied to a test problem for the heat equation with Stefan-Boltzmann boundary condition. The numerical tests confirm the theoretical results.     

Pontryagin’s Principle for Optimal Control Problem Governed by 3D Navier–Stokes Equations

This paper deals with the Pontryagin maximum principle for optimal control problems governed by 3D Navier–Stokes equations with pointwise control constraint. The obtained result is proved by using some results on regularity of solutions of the Navier–Stokes equations and techniques of optimal control theory.     

On the Non-Validity of the Order Reduction Method for Singularly Perturbed Control Systems

The order reduction method for singularly perturbed optimal control systems consists of setting the small parameter equal to zero and employing the differential system thus obtained. Although in many situations this provides the correct variational limit problem, it is established in this paper that when considering systems with non-scalar fast variables, the set of systems for which the order reduction method is invalid is dense in the class of systems under consideration. This extends previous results, where only systems with linear fast variables were considered. The present result complements a result established in a joint work with Artstein, where it was established that the order reduction method is valid for singularly perturbed optimal control systems with scalar fast variable.     

Analysis and numerical simulation of a nonlinear mathematical model for testing the manoeuvrability capabilities of a submarine

The aim of this work is to provide a mathematical and numerical tool for the analysis of the manoeuvrability capabilities of a submarine. To this end, we consider a suitable optimal control problem with constraints in both state and control variables. The state law is composed of a highly coupled and nonlinear system of twelve ordinary differential equations. Control inputs appear in linear and quadratic form and physically are linked to rudders and propeller forces and moments. We consider a nonlinear Bolza type cost function which represents a commitment between reaching a final desired state and a minimal expense of control. In a first part, following recent ideas in [F. Periago, J. Tiago, A local existence result for an optimal control problem modeling the manoeuvring of an underwater vehicle, Nonlinear Anal. RWA 11 (2010) 2573–2583], we prove a local existence result for the above mentioned optimal control problem. In a second part, we address the numerical resolution of the problem by using a descent method with projection and optimal step-size parameter. To illustrate the performance of the method proposed in this paper and to show its application in a real engineering problem we include three different numerical experiments for a standard manoeuvre.     

Optimal institutional advertising: Minimum-time problem

This paper considers the problem of optimizing the institutional advertising expenditure for a firm which produces two products. The problem is formulated as a minimum-time control problem for the dynamics of an extended Vidale-Wolfe advertising model, the optimal control being the rate of institutional advertising that minimizes the time to attain the specified target market shares for the two products. The attainable set and the optimal control are obtained by applying the recent theory developed by Hermes and Haynes extending the Green's theorem approach to higher dimensions. It is shown that the optimal control is a strict bang-bang control. An interesting side result is that the singular arc obtained by the Green's theorem application turns out to be a maximum-time solution over the set of all feasible controls. The result clarifies the connection between the Green's theorem approach and the maximum principle approach.     

Maximum Principle for Partially-Observed Optimal Control of Fully-Coupled Forward-Backward Stochastic Systems

This paper is concerned with partially-observed optimal control problems for fully-coupled forward-backward stochastic systems. The maximum principle is obtained on the assumption that the forward diffusion coefficient does not contain the control variable and the control domain is not necessarily convex. By a classical spike variational method and a filtering technique, the related adjoint processes are characterized as solutions to forward-backward stochastic differential equations in finite-dimensional spaces. Then, our theoretical result is applied to study a partially-observed linear-quadratic optimal control problem for a fully-coupled forward-backward stochastic system and an explicit observable control variable is given.     

Stochastic Maximum Principle for Forward-Backward Regime Switching Jump Diffusion Systems and Applications to Finance

The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The result is applied to a cash flow valuation problem with terminal wealth constraint in a financial market. An explicit optimal strategy is obtained in this example.