One control problem of a pencil of trajectories

A linear control problemwith a nonsingleton initial set is dealt with. For this problem,we consider the problem of transforming this initial set along trajectories of a linear controlled system into some fixed terminal set in a finite time using a single control. Various aspects of this control problem are studied using the machinery of support functions.     

Approximate controllability of impulsive fractional evolution equations of order 1<α<2$$ 1&amp;lt;\alpha &amp;lt;2 $$ with state-dependent delay in Banach spaces

This article deals with the approximate controllability problem for fractional evolution equations involving noninstantaneous impulses and state-dependent delay. In order to derive sufficient conditions for the approximate controllability of our problem, we first consider the linear-regulator problem and find the optimal control in the feedback form. By using this optimal control, we develop the approximate controllability of the linear fractional control system. Further, we obtain sufficient conditions for the approximate controllability of the nonlinear problem. In the end, we provide a concrete example to support the applicability of the derived results.     

Geometric control of hybrid systems

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support functions.     

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.     

An approach for solving of a moving boundary problem

In this paper we shall study moving boundary problems, and we introduce an approach for solving a wide range of them by using calculus of variations and optimization. First, we transform the problem equivalently into an optimal control problem by defining an objective function and artificial control functions. By using measure theory, the new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; then we obtain an optimal measure which is then approximated by a finite combination of atomic measures and the problem converted to an infinite-dimensional linear programming. We approximate the infinite linear programming to a finite-dimensional linear programming. Then by using the solution of the latter problem we obtain an approximate solution for moving boundary function on specific time. Furthermore, we show the path of moving boundary from initial state to final state.     

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.     

Linear-quadratic control and quadratic differential forms for multidimensional behaviors

This paper deals with systems described by constant coefficient linear partial differential equations (nD-systems) from a behavioral point of view. In this context we treat the linear quadratic control problem where the performance functional is the integral of a quadratic differential form. We look for characterizations of the set of stationary trajectories and of the set of local minimal trajectories with respect to compact support variations, turning out that they are equal if the system is dissipative. Finally we provide conditions for regular implementability of this set of trajectories and give an explicit representation of an optimal controller.     

A discontinuous interpolated finite volume approximation of semilinear elliptic optimal control problems

In this article, we describe a discontinuous finite volume method with interpolated coefficients for the numerical approximation of the distributed optimal control problem governed by a class of semilinear elliptic equations with control constraints. The proposed distributed control problem involves three unknown variable: control, state and costate. For the approximation of control, we have adopted three different methodologies: variational discretization, piecewise constant and piecewise linear discretization, while the approximation of state and costate variables is based on discontinuous piecewise linear polynomials. As the resulted scheme is non‐symmetric, optimize‐then‐discretize approach is used to approximate the control problem. Optimal a priori error estimates in suitable natural norms for state, costate and control variables are derived. Moreover, numerical experiments are presented to support the derived theoretical results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2090–2113, 2017     

An LP approach to dynamic programming principles for stochastic control problems with state constraints

We study a class of nonlinear stochastic control problems with semicontinuous cost and state constraints using a linear programming (LP) approach. First, we provide a primal linearized problem stated on an appropriate space of probability measures with support contained in the set of constraints. This space is completely characterized by the coefficients of the control system. Second, we prove a semigroup property for this set of probability measures appearing in the definition of the primal value function. This leads to dynamic programming principles for control problems under state constraints with general (bounded) costs. A further linearized DPP is obtained for lower semicontinuous costs.     

Solving the maximum problem for the terminal norm of a linear control system

We consider the maximization problem for the terminal norm of a linear control system. We seek for the extremal controls and improve them. Based on a sufficient optimality condition, we construct procedures that use the differentiability of the support function of the set of attainability.     

An approach for solving nonlinear programming problems

In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures; then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.     

An Indefinite Stochastic Linear Quadratic Optimal Control Problem with Delay and Related Forward–Backward Stochastic Differential Equations

In this paper, we will study an indefinite stochastic linear quadratic optimal control problem, where the controlled system is described by a stochastic differential equation with delay. By introducing the relaxed compensator as a novel method, we obtain the well-posedness of this linear quadratic problem for indefinite case. And then, we discuss the uniqueness and existence of the solutions for a kind of anticipated forward–backward stochastic differential delayed equations. Based on this, we derive the solvability of the corresponding stochastic Hamiltonian systems, and give the explicit representation of the optimal control for the linear quadratic problem with delay in an open-loop form. The theoretical results are validated as well on the control problems of engineering and economics under indefinite condition.     

A new approach for asymptotic stability system of the nonlinear ordinary differential equations

In this paper, we use measure theory for considering asymptotically stable of an autonomous system [1] of first order nonlinear ordinary differential equations(ODE’s). First, we define a nonlinear infinite-horizon optimal control problem related to the ODE. Then, by a suitable change of variable, we transform the problem to a finite-horizon nonlinear optimal control problem. Then, the problem is modified into one consisting of the minimization of a linear functional over a set of positive Radon measures. The optimal measure is approximated by a finite combination of atomic measures and the problem converted to a finite-dimensional linear programming problem. The solution to this linear programming problem is used to find a piecewise-constant control, and by using the approximated control signals, we obtain the approximate trajectories and the error functional related to it. Finally the approximated trajectories and error functional is used to for considering asymptotically stable of the original problem.     

Vibrational stabilization and the Brockett problem

The present paper deals with the exposition of methods for solving the Brockett problem on the stabilization of linear control systems by a nonstationary feedback. The paper consists of two parts. We consider continuous linear control systems in the first part and discrete systems in the second part. In the first part, we consider two approaches to the solution of the Brockett problem. The first approach permits one to obtain low-frequency stabilization, and the second part deals with high-frequency stabilization. Both approaches permit one to derive necessary and sufficient stabilization conditions for two-dimensional (and three-dimensional, for the first approach) linear systems with scalar inputs and outputs. In the second part, we consider an analog of the Brockett problem for discrete linear control systems. Sufficient conditions for low-frequency stabilization of linear discrete systems are obtained with the use of a piecewise constant periodic feedback with sufficiently large period. We obtain necessary and sufficient conditions for the stabilization of two-dimensional discrete systems. In the second part, we also consider the control problem for the spectrum (the pole assignment problem) of the monodromy matrix for discrete systems with a periodic feedback.     

Optimal control of the heat equation in an inhomogeneous body

In this paper we consider a heat flow in an inhomogeneous body without internal source. There exists special initial and boundary conditions in this system and we intend to find a convenient coefficient of heat conduction for this body so that body cool off as much as possible after definite time. We consider this problem in a general form as an optimal control problem which coefficient of heat conduction is optimal function. Then we replace this problem by another in which we seek to minimize a linear form over a subset of the product of two measures space defined by linear equalities. Then we construct an approximately optimal control.     

Approximate solution of a control problem on the basis of the nilpotent approximation

We consider the control problem for deterministic systems described by ordinary differential equations with linear controls. On the basis of the nilpotent approximation method, we construct an algorithm for finding an approximate solution of the control problem for three-dimensional nonlinear systems with two linear controls. The algorithm was implemented in Maple and tested in examples including the control of a mobile robot on a plane and the attitude control of a sphere rolling on a plane.     

Asymptotic solution method for a quasilinear minimum force control problem

For a transient process in a quasilinear system, we consider an optimization problem of finding a (multi-dimensional) control with minimum intensity. We suggest an algorithm for constructing asymptotic approximations to the solution of this problem. The main advantage of the algorithm is that an optimal control problem for a linear system is solved instead of the original essentially nonlinear problem.     

An Open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information: Linear control systems

The method of open-loop control packages is a tool for stating the solvability of guaranteed closed-loop control problems under incomplete information on the observed states. In this paper, a solution method is specified for the problem of guaranteed closed-loop guidance of a linear control system to a convex target set at a prescribed point in time. It is assumed that the observed signal on the system’s states is linear and the set of its admissible initial states is finite. It is proved that the problem under consideration is equivalent to the problem of open-loop guidance of an extended linear control system to an extended convex target set. Using a separation theorem for convex sets, we derive a solvability criterion, which reduces to solving a finite-dimensional optimization problem. An illustrative example is considered.     

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.