Method of penalization for the state equation for an elliptical optimal control problem

We solve by finite difference method an optimal control problem of a system governed by a linear elliptic equation with pointwise control constraints and non-local state constraints. A discrete optimal control problem is approximated by a minimization problem with penalized state equation. We derive the error estimates for the distance between the exact and regularized solutions. We also prove the rate of convergence of block Gauss–Seidel iterative solution method for the penalized problem. We present and analyze the results of the numerical experiments.     

Bernoulli substitution in the Ramsey model: Optimal trajectories under control constraints

We consider a neoclassical (economic) growth model. A nonlinear Ramsey equation, modeling capital dynamics, in the case of Cobb-Douglas production function is reduced to the linear differential equation via a Bernoulli substitution. This considerably facilitates the search for a solution to 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. A proposed algorithm for constructing growth trajectories combines methods of open-loop control and closed-loop regulatory control. For some levels of constraints and initial conditions, a closed-form solution is obtained. We also demonstrate the impact of technological change on the economic equilibrium dynamics. Results are supported by computer calculations.     

Optimal control problem with an integral equation as the control object

We consider a nonlinear optimal control problem with an integral equation as the control object, subject to control constraints. This integral equation corresponds to the fractional moment of a stochastic process involving short-range and long-range dependences. For both cases, we derive the first-order necessary optimality conditions in the form of the Euler–Lagrange equation, and then apply them to obtain a numerical solution of the problem of optimal portfolio selection.     

Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible control is mean-field type.Making full use of the backward stochastic differential equation theory,we transform the original control system into an equivalent backward form,i.e.,the equations in the control system are all backward.In addition,Ekeland’s variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control.We also study a stochastic linear quadratic control problem with state constraints.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

Stochastic Optimal Control and Linear Programming Approach

We study a classical stochastic optimal control problem with constraints and discounted payoff in an infinite horizon setting. The main result of the present paper lies in the fact that this optimal control problem is shown to have the same value as a linear optimization problem stated on some appropriate space of probability measures. This enables one to derive a dual formulation that appears to be strongly connected to the notion of (viscosity sub) solution to a suitable Hamilton-Jacobi-Bellman equation. We also discuss relation with long-time average problems.     

On unbounded solutions of Bellman's equation associated with optimal switching control problems with state constraints

We study a quasi-variational inequality system with unbounded solutions. It represents the Bellman equation associated with an optimal switching control problem with state constraints arising from production engineering. We show that the optimal cost is the unique viscosity solution of the system.This work was supported by the National Research Council of Argentina, Grant No. PID-BID 213.     

Approximate Penalty Method in Optimal Control Problems for Nonsmooth Singular Systems

We consider an abstract optimal control problem with additional constraints and nonsmooth terms, but without the requirement that both the state equation on the set of admissible controls and the extremum problem be solvable. We use the approximate penalty method proposed here to find an approximate (in the weak sense) solution of the problem. As an example, we consider the optimal control problem for a singular nonlinear elliptic type equation.     

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.     

Optimization for nonlinear hyperbolic equations without the uniqueness theorem for a solution of the boundary-value problem

We consider the optimal control problem for a system governed by a nonlinear hyperbolic equation without any constraints on the parameter of nonlinearity. No uniqueness theorem is established for a solution to this problem. The control-state mapping of this system is not Gateaux differentiable. We study an approximate solution of the optimal control problem by means of the penalty method.     

Solution of elliptic optimal control problem with pointwise and non-local state constraints

We study an optimal control problem of a system governed by a linear elliptic equation, with pointwise control constraints and pointwise and non-local (integral) state constraints. We construct a finite-difference approximation of the problem, we prove the existence and the convergence of the approximate solutions to the exact solution. We construct and study mesh saddle point problem and its iterative solution method and analyze the results of numerical experiments.     

Strong controllability and optimal control of the heat equation with a thermal source

In this paper we consider an optimal control system described byn-dimensional heat equation with a thermal source. Thus problem is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. Here we assume there is no constraints on control. This problem is reduced to a moment problem.We modify the moment problem into one consisting of the minimization of a positive linear functional over a set of Radon measures and we show that there is an optimal measure corresponding to the optimal control. The above optimal measure approximated by a finite combination of atomic measures. This construction gives rise to a finite dimensional linear programming problem, where its solution can be used to determine the optimal combination of atomic measures. Then by using the solution of the above linear programming problem we find a piecewise-constant optimal control function which is an approximate control for the original optimal control problem. Finally we obtain piecewise-constant optimal control for two examples of heat equations with a thermal source in one-dimensional.     

Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem

This paper concerns optimal investment problem of a CRRA investor who faces proportional transaction costs and finite time horizon. From the angle of stochastic control, it is a singular control problem, whose value function is governed by a time-dependent HJB equation with gradient constraints. We reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. This enables us to make use of the well-developed theory of obstacle problem to attack the problem. The C2,1 regularity of the value function is proven and the behaviors of the free boundaries are completely characterized.     

A complete asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with geometric constraints

We consider an optimal control problem for solutions of a boundary value problem on an interval for a second-order ordinary differential equation with a small parameter at the second derivative. The control is scalar and is subject to geometric constraints. Expansions of a solution to this problem up to any power of the small parameter are constructed and justified.     

Relaxation in nonconvex optimal control problems described by fractional differential equations

We consider the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a control system described by fractional differential equation with mixed nonconvex constraints on the control. A relaxation problem is treated along with the original problem. It is proved that, under general assumptions, the relaxation problem has an optimal solution, and that for each optimal solution there is a minimizing sequence of the original problem that converges to the optimal solution with respect to the trajectory, the control, and the functional in appropriate topologies simultaneously.     

Ergodic control problems for optimal stochastic production planning with production constraints

We study the ergodic control problem related to stochastic production planning in a single product manufacturing system with production constraints. The existence of a solution to the corresponding Hamilton-Jacobi-Bellman equation and its properties are shown. Furthermore, the optimal control for the ergodic control problem and an example are given.     

Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints

This paper deals with the optimal control problem of an ordinary differential equation with several pure state constraints, of arbitrary orders, as well as mixed control-state constraints. We assume (i) the control to be continuous and the strengthened Legendre–Clebsch condition to hold, and (ii) a linear independence condition of the active constraints at their respective order to hold. We give a complete analysis of the smoothness and junction conditions of the control and of the constraints multipliers. This allows us to obtain, when there are finitely many nontangential junction points, a theory of no-gap second-order optimality conditions and a characterization of the well-posedness of the shooting algorithm. These results generalize those obtained in the case of a scalar-valued state constraint and a scalar-valued control.     

A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.     

A priori error estimates for elliptic optimal control problems with a bilinear state equation

In this paper a priori error analysis for the finite element discretization of an optimal control problem governed by an elliptic state equation is considered. The control variable enters the state equation as a coefficient and is subject to pointwise inequality constraints. We derive a priori error estimates for the discretization error in the control variable and confirm our theoretical results by numerical examples.     

具有点态控制约束热方程的时间与范数最优控制问题的等价性

本文研究了具有点态控制热方程的等价性问题.利用变分法分析时间最优控制的唯一性,能控性以及范数最优控制的特征,获得了具有点态控制约束热方程的时间与范数最优控制问题之间的等价性,推广了现有文献的结果.