An Artificial Neural Network for Solving Distributed Optimal Control of the Poisson’s Equation

This paper focuses on addressing the linear quadratic (LQ) optimal control problem on an infinite time horizon for stochastic systems controlled by impulses. No constraint regarding the sign of the quadratic functional is applied. That is why our first concern is to find conditions which guarantee that the considered optimal control problem is well posed. Then, when the optimal control problem is well posed, it is natural to look for conditions which guarantee the attainability of the optimal control problem that is being evaluated. The main tool involved in the solution of the problems stated before is a backward jump matrix linear differential equation (BJMLDE) with a Riccati-type jumping operator. This is formulated using the matrix coefficients of the controlled system and the weight matrices of the performance criterion. We show that the well posedness of the optimal control problem under investigation is guaranteed by the existence of the maximal and bounded solution of the associated BJMLDE with a Riccati-type jumping operator. Further, we show that when the associated BJMLDE with a Riccati-type jumping operator has a maximal solution which satisfies a suitable sign condition, then the optimal control problem is attainable if and only if it has an optimal control in a state feedback form, or if and only if the maximal solution of the BJMLDE with a Riccati-type jumping operator is a stabilizing solution. In order to make the paper more self-contained, we present a set of conditions that correspond to the existence of the maximal solution of the BJMLDE satisfying the desired sign condition.

In this article, we consider a linear-quadratic optimal control problem (LQ problem) for a controlled linear stochastic differential equation driven by a multidimensional Browinan motion and a Poisson random martingale measure in the general case, where the coefficients are allowed to be predictable processes or random matrices. By the duality technique, the dual characterization of the optimal control is derived by the optimality system (so-called stochastic Hamilton system), which turns out to be a linear fully coupled forward-backward stochastic differential equation with jumps. Using a decoupling technique, the connection between the stochastic Hamilton system and the associated Riccati equation is established. As a result, the state feedback representation is obtained for the optimal control. As the coefficients for the LQ problem are random, here, the associated Riccati equation is a highly nonlinear backward stochastic differential equation (BSDE) with jumps, where the generator depends on the unknown variables K, L, and H in a quadratic way (see (5.9) herein). For the case where the generator is bounded and is linearly dependent on the unknown martingale terms L and H, the existence and uniqueness of the solution for the associated Riccati equation are established by Bellman's principle of quasi-linearization.

On the regularity of solutions of an operator Riccati equation arising in linear quadratic optimal control problems for hereditary differential systems

We propose a new dual projected pseudo quasi-Newton method for discrete-time linear quadratic optimal control problems of a large-scale multi-destination network. Our method has successfully overcome the difficulties caused by large dimensions and inequality constraints by having a constant Hessian matrix and an efficient successive projection method, respectively. Application of our method to the congestion control of multi-destination networks is presented. Above all, it takes less than one second to obtain the solution of a large-scale discrete-time linear quadratic optimal control problem with the problem size of 2800 variables, 700 equality constraints and 2400 inequality constraints.

On the stochastic linear quadratic control problem with piecewise constant admissible controls

This paper extends the generating function approach to the state constrained linear quadratic optimal control problem by utilizing penalties. The original constrained problem is approximated by an unconstrained problem with a novel penalty function by which the latter problem can be solved via generating function method based on Taylor series approximation. In view of this, the optimal solution can be given as the state feedback control with explicit boundary conditions and pre-computed coefficients. This structure is useful in the repetitive on-line computation for optimal solutions satisfying a number of different boundary conditions. Examples demonstrate the effectiveness of the developed method.

In this paper, we use the solutions of forward-backward stochastic differential equations with random jumps to get the optimal controls for forward-backward stochastic linear quadratic optimal control problem and the Nash equilibrium point for nonzero sum differential games problem.

Linear Quadratic Optimal Control of a Wave Equation with Boundary Damping and Pointwise Control Input

A stochastic control problem is formulated and we get the explicit form of the optimal control for initial-data-parameterized linear quadratic stochastic optimal control problems with random jumps. The optimal control can be proved to be unique. A stochastic Riccati equation is rigorous derived from the stochastic Hamilton system, which provides an optimal feedback control. This completes the the interrelationship between the stochastic Riccati equation and stochastic Hamilton system as two different but equivalent tools for the stochastic linear quadratic problem.

This communication presents a spectral method for solving time-varying linear quadratic optimal control problems. Legendre–Gauss–Lobatto nodes are used to construct the mth-degree polynomial approximation of the state and control variables. The derivative x·(t) of the state vector x(t) is approximaed by the analytic derivative of the corresponding interpolating polynomial. The performance index approximation is based on Gauss–Lobatto integration. The optimal control problem is then transformed into a linear programming problem. The proposed technique is easy to implement, efficient and yields accurate results. Numerical examples are included and a comparison is made with an existing result.

In this paper, we formulate and investigate a framework for the theory of the linear quadratic optimal control problem (LQ problem) for infinite‐dimensional mean‐field stochastic evolution systems with jumps. We ensure the well‐posedness of the investigated problems by establishing the existence, uniqueness, and a priori estimates for mild solutions to general infinite‐dimensional mean‐field forward stochastic evolution equations (MF‐SEE) and mean‐field backward stochastic evolution equations (MF‐BSEE) with jumps. Leveraging the Yosida approximation theory, we establish a dual theory between MF‐SEE and MF‐BSEE with jumps, overcoming the challenge posed by the inapplicability of Itô's formula in the context of mild solutions. Our main results regarding the existence and uniqueness of the optimal control, along with corresponding dual characterizations and state feedback representations, are obtained through convex analysis techniques, our established dual theory, and decoupling methods.

The linear quadratic optimal control problem under quadratic constraints is an optimization problem over a generally non-convex set. Yakubovich [8] and Megretski [4] have studied this problem, and they show how it may be translated into a two-stage, convex optimization problem. In this paper we study the linear quadratic control problem under quadratic constraints for generalized first-order systems. As in the state-space case the linear quadratic control problem without quadratic constraints may be solved in terms of a linear matrix inequality. Subsequently, we use the results from [8] to derive a linear matrix inequality characterizing the linear quadratic optimal behaviour under quadratic constraints.

This paper studies the discrete-time indefinite stochastic linear quadratic optimal control problem with the equality terminal state constraints,which can be transformed into a mathematical programming with equality constraints. By means of matrix Lagrange theorem, a necessary condition for the existence of optimal linear state feedback control is given. The previous results on discrete-time stochastic linear quadratic optimal control without constraints, can be viewed as corollaries of the main theorem of this paper.

This paper deals with feedback control over a single fixed-rate channel using predictive coding at the transmitter side. The central thrust is to demonstrate that optimal control based on predictive coding plus fixed memoryless quantization at the transmitter, designed to improve the efficiency of the channel usage and exemplified (or perhaps extremized) by the transmission of the quantized innovations signal, in general, requires the construction of the joint density of both the plant and predictor states at the receiver side and inherits a plant stability requirement, which is examined. The Bayesian filter is developed. This recursive filter's state density is used to compute the optimal feedback control. This is in contrast to the less complicated propagation solely of the predictor state, which would suffice in the linear quadratic optimal control problem-a feature that is elucidated. A linear nonquadratic optimal control example is provided to illustrate the approach and its benefits over control based on the recovered predictor state density or control without predictive coding. In each of these competing cases, a lower complexity receiver architecture is possible but at the expense of closed-loop control performance.

In this paper, an iterative computational method is proposed to handle constrained nonlinear quadratic optimal control problems. with terminal state constraints, saturation state and saturation control variables constraints. The method is based on replacing the original constrained nonlinear optimal control problem by a sequence of constrained linear time-varying quadratic optimal control problems. To this end, an iterative technique is used to replace the original constrained nonlinear dynamic system by a sequence of constrained linear time-varying systems. Then each of constrained linear time-varying quadratic optimal control problems is approximated by a quadratic programming problem by parameterizing each of the state variable by a finite length Legendre polynomials with unknown parameters. To show the effectiveness of the proposed method, simulation results of a constrained nonlinear optimal control problem are presented.