The existence results for optimal control problems governed by a variational inequality

In this paper, we establish the existence of the optimal control for an optimal control problem where the state of the system is defined by a variational inequality problem with monotone type mappings. Moreover, as an application, we get several existence results of an optimal control for the optimal control problem where the system is defined by a quasilinear elliptic variational inequality problem with an obstacle.     

Duality of linear estimation and control with constraints

This paper considers the duality of linear estimation and control with constraints. Specifically, the dual of the optimal filter, where the mean-squared values of the optimal estimates are constrained, is shown to be an optimal control problem with constraints on the control, and the dual of the estimation of stochastic signals in the presence of multiplicative noise in addition to measurement noise is shown to be a control problem with constraints on the states, with the boundary conditions specified at opposite ends of the extremals in time. Algorithms are presented to establish the duality concept.     

Inverse problem for the reaction diffusion system by optimization method

In this paper, we study an inverse problem of reconstructing two time independent coefficients in the reaction diffusion system from the final measurement. First the given problem is transformed into an optimization problem by using optimal control framework and the existence of the minimizer for the control functional is established. Then we prove the stability estimate for two coefficients with the upper bound given by some Sobolev norms of the final measurement.     

Optimal control of a two-dimensional contact problem

We consider a frictionless contact problem with unilateral constraints for a 2D bar. We describe the problem, then we derive its weak formulation, which is in the form of an elliptic variational inequality of the first kind. Next, we establish the existence of a unique weak solution to the problem and prove its continuous dependence with respect to the applied tractions and constraints. We proceed with the study of an associated control problem for which we prove the existence of an optimal pair. Finally, we consider a perturbed optimal control problem for which we prove a convergence result.     

On the Existence of Solutions of Two Optimization Problems

In this paper, we prove the existence of solutions for the minimization problem of the shell weight for a given minimal frequency of the shell vibrations as well as for the maximization problem of the minimal frequency for a given shell weight. We consider an optimal control problem governed by an eigenvalue problem for a system of differential equations with variable coefficients. The form of the shell is considered as a control. Some of the coefficients are non-measurable. Earlier, we introduced certain special weighted functional spaces. By using these spaces, we establish the continuity of the considered minimal frequency functional and obtain the existence of solutions of both optimal control problems. At the end, we prove the Lipschitz continuity of the eigenvalue problem.     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

Risk sensitive impulse control of non-Markovian processes

We consider the problem of an optimal stochastic impulse control of non-Markovian Processes when the expression of the cost functional integrates sensitiveness with respect to the risk. For this class, we try to establish the existence of an optimal strategy. We prove that our impulse control problem could be reduced to an iterative sequence of optimal stopping ones. Basically, the problem is solved using techniques involving the Snell envelope notion.     

Carleman estimate for parabolic equation with memory effects and exact controllability

A nonlinear parabolic problem with memory effect is considered. We first establish a Carleman type estimate for a linear problem which is the adjoint of a suitable linearization of the nonlinear problem. As a consequence of this estimate we obtain an observability estimate. Then we establish the exact controllability of the linearized system with distributed control over a subdomain. Finally, we arrive at the controllability of the nonlinear system via a classical fixed point theorem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A maximum principle for fully coupled stochastic control systems of mean-field type

The present paper considers an optimal control problem for fully coupled forward–backward stochastic differential equations (FBSDEs) of mean-field type in the case of controlled diffusion coefficient. Moreover, the control domain is not assumed to be convex. By virtue of a reduction method, we establish the necessary optimality conditions of Pontryagin's type. As an application, a linear–quadratic stochastic control problem is studied.     

Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs

We consider an optimal impulse control problem on reinsurance, dividend and reinvestment of an insurance company. To close reality, we add fixed and proportional transaction costs to this problem. The value of the company is associated with expected present value of net dividends pay out minus the net reinvestment capitals until ruin time. We focus on non-cheap proportional reinsurance. We prove that the value function is a unique solution to associated Hamilton–Jacobi–Bellman equation, and establish the regularity property of the viscosity solution under a weak assumption. We solve the non-uniformly elliptic equation associated with the impulse control problem. Finally, we derive the value function and the optimal strategy of the control problem.     

Optimal control for an age-dependent predator-prey system in a polluted environment

In this paper, we investigate an optimal control problem for an age-dependent predator-prey hybrid system in a polluted environment. Namely, we establish a new toxicant-population model with age-dependence in an environment with small toxicant capacity. By using a fixed point theorem, we obtain the existence and uniqueness of nonnegative solution of the toxicant-population model with age-dependence. We employ tangent-normal cone techniques in nonlinear functional analysis to deduct the optimality conditions for the control problem. The existence of optimal control policy is carefully verified by means of Ekeland’s variational principle.     

Existence of solutions and periodic solutions for nonlinear evolution inclusions

In this paper we consider nonlinear-dependent systems with multivalued perturbations in the framework of an evolution triple of spaces. First we prove a surjectivity result for generalized pseudomonotone operators and then we establish two existence theorems: the first for a periodic problem and the second for a Cauchy problem. As applications we work out in detail a periodic nonlinear parabolic partial differential equation and an optimal control problem for a system driven by a nonlinear parabolic equation.     

Simultaneous Identification of Two Time Independent Coefficients in a Nonlinear Phase Field System

In this article, we study a simultaneous reconstruction of two time independent parameters in a nonlinear phase field system by final overdetermination data. To this end, the given problem is transformed into an optimization problem by using the optimal control framework; then the existence of the minimizer for the control functional is established. Further, we deduce the necessary condition for the minimizer of the control functional. Finally we derive the stability estimate for two coefficients with the upper bound given by some Sobolev norms of the final measurement of the solutions.     

Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls

Summary. An optimal control problem for impressed cathodic systems in electrochemistry is studied. The control in this problem is the current density on the anode. A matching objective functional is considered. We first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition. We then prove the existence of an optimal solution. Next, we derive a necessary condition of optimality and establish an optimality system of equations. Finally, we define a finite element algorithm and derive optimal error estimates. Received March 10, 1993 / Revised version received July 4, 1994     

Evasion with a single measurement of the state vector

We consider a plant whose dynamics is described by a system of linear ordinary differential equations with an unknown noise. The considered differential game has fixed terminal time. The players’ control parameters are subjected to geometric constraints in the form of closed balls. The aim of the control is to evade zero. The problem is solved with the use of control laws with a single measurement of the state vector. We present necessary conditions and a construction algorithm for the optimal measurement instant. The results are illustrated by some control systems of a special form. In particular, we consider an example in which one measurement can guarantee the result not worse than that for a refined grid of measurement points. We discuss whether this situation is typical. We also consider an example with infinitely many points at which the unique measurement can be performed.     

On the existence of optimal controls for nonlinear infinite dimensional systems

We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations.     

Optimal Control of a Stochastic Hybrid System with Discounted Cost

We address the optimal control problem of a very general stochastic hybrid system with both autonomous and impulsive jumps. The planning horizon is infinite and we use the discounted-cost criterion for performance evaluation. Under certain assumptions, we show the existence of an optimal control. We then derive the quasivariational inequalities satisfied by the value function and establish well-posedness. Finally, we prove the usual verification theorem of dynamic programming.     

Optimal control of impulsive systems with nonlocal boundary conditions

In this paper we study an optimal control problem, where states of a control system are described by impulsive differential equations with nonlocal boundary conditions. With the help of the contraction principle we prove the existence and uniqueness of a solution to the corresponding boundary value problem with fixed admissible controls. We calculate the first and second variation of the functional. Using the variation of controls, we establish various necessary optimality conditions of the second order.     

Optimal control of Maxwell’s equations with regularized state constraints

This paper is devoted to an optimal control problem of Maxwell??s equations in the presence of pointwise state constraints. The control is given by a divergence-free three-dimensional vector function representing an applied current density. To cope with the divergence-free constraint on the control, we consider a vector potential ansatz. Due to the lack of regularity of the control-to-state mapping, existence of Lagrange multipliers cannot be guaranteed. We regularize the optimal control problem by penalizing the pointwise state constraints. Optimality conditions for the regularized problem can be derived straightforwardly. It also turns out that the solution of the regularized problem enjoys higher regularity which then allows us to establish its convergence towards the solution of the unregularized problem. The second part of the paper focuses on the numerical analysis of the regularized optimal control problem. Here the state and the control are discretized by Nédélec??s curl-conforming edge elements. Employing the higher regularity property of the optimal control, we establish an a priori error estimate for the discretization error in the $\boldsymbol{H}(\bold{curl})$ -norm. The paper ends by numerical results including a numerical verification of our theoretical results.     

Procedure for asymptotic state and parameter estimation of nonlinear distributed parameter bioreactors

Due to measurement problem in biological variables, such as the biomass concentration and its specific growth rate, the on-line measurement of these variables is not ensured and this presents a great difficulty, especially when we consider the control problem. This paper presents and analyses an approach for estimating these biological variables through an example of a nonlinear distributed parameter bioreactor. An orthogonal collocation method is applied in the distributed system to obtain a system of ordinary differential equations. A simulation study shows the feasibility and the robustness of the estimator used for this nonlinear process.