On the generalized algebraic Riccati equations

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.

ABSTRACTA finite horizon linear quadratic (LQ) optimal control problem is studied for a class of discrete-time linear fractional systems (LFSs) affected by multiplicative, independent random perturbations. Based on the dynamic programming technique, two methods are proposed for solving this problem. The first one seems to be new and uses a linear, expanded-state model of the LFS. The LQ optimal control problem reduces to a similar one for stochastic linear systems and the solution is obtained by solving Riccati equations. The second method appeals to the principle of optimality and provides an algorithm for the computation of the optimal control and cost by using directly the fractional system. As expected, in both cases, the optimal control is a linear function in the state and can be computed by a computer program. A numerical example and comparative simulations of the optimal trajectory prove the effectiveness of the two methods. Some other simulations are obtained for different values of the fractional order.

A Class of Linear Quadratic Gaussian Hybrid Optimal Control Problems with Realization–Independent Riccati Equations

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.

SummaryIn this article, we study the open‐loop and closed‐loop solvability for indefinite mean‐field stochastic linear quadratic (MF‐SLQ) optimal control problem and its application in finance, where the controlled stochastic system is driven by a Brownian motion and a Poisson random martingale measure and also disturbed by some stochastic processes. The intrinsic property of stochastic systems results in the inequivalence of those two solvabilities, which is different from deterministic case. Based on a well‐posedness result of the problem, it is shown that the uniform convexity of cost functional is sufficient for the open‐loop solvability of the problem. By a matrix minimum principle, the necessity condition of regular solvability for a decoupled Riccati equation is established, meanwhile, the closed‐loop solvability is turned out to be equivalent to the regular solvability of Riccati equations with some constraints on the solutions of disturbances equations, moreover, the optimal closed‐loop strategy is characterized by regular solutions of Riccati equations and adapted solutions of disturbances equations. And then a mean‐variance model is considered to solve optimal portfolio selection strategy problem of the insurance company with liability. Our study fills a gap in the field of solvability for mean‐field stochastic optimal control with random jumps and provides a different way for solving mean‐variance portfolio selection model.

In this paper, we consider optimal control problems of stochastic systems with delay. The control domains are not necessarily convex and the diffusion coefficients explicitly depend on control variables. The delay term of the state is regarded as a control process and then the Ekeland's variational principle is applied to obtain a general maximum principle for the optimal control problems. As applications, we use the theoretical results to a delayed stochastic linear quadratic (LQ) optimal control problem with a non-convex control domain.

Duality between estimation and optimal control is a problem of rich historical significance. The first duality principle appears in the seminal paper of Kalman-Bucy, where the problem of minimum variance estimation is shown to be dual to a linear quadratic (LQ) optimal control problem. Duality offers a constructive proof technique to derive the Kalman filter equation from the optimal control solution.This paper generalizes the classical duality result of Kalman-Bucy to the nonlinear filter: The state evolves as a continuous-time Markov process and the observation is a nonlinear function of state corrupted by an additive Gaussian noise. A dual process is introduced as a backward stochastic differential equation (BSDE). The process is used to transform the problem of minimum variance estimation into an optimal control problem. Its solution is obtained from an application of the maximum principle, and subsequently used to derive the equation of the nonlinear filter. The classical duality result of Kalman-Bucy is shown to be a special case.

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.

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.

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.

A framework is presented for the destabilization of dynamical systems that are subject to disturbances and controls. Such destabilizing controllers are designed using a methodology analogous to gain scheduling. At each point in the state space, the controller is designed to destabilize the local linearization of the system. The local destabilization objective is formulated as a linear quadratic (LQ) optimal control problem with an indefinite performance objective, and is therefore similar to problems in differential games and H/sup /spl infin// norm calculations. Optimal feedback controls are thus readily computed by solving the appropriate Ricatti equations. This problem formulation is motivated by the objective of mixing enhancement in nonlinear dynamical systems, which in turn motivates the consideration of these LQ problems with finite-time horizons and exponentially discounted performance objectives.

A new algorithmic setting is proposed for the discrete-time finite-horizon linear quadratic (LQ) optimal control problem with constrained or unconstrained final state, no matter whether the problem is cheap, singular, or regular. The proposed solution, based on matrix pseudoinversion, is completed and made practically implementable by a nesting procedure for welding optimal subarcs that enables arbitrary enlargement of the control time interval.

Based on continuous Hopfield neural network (CHNN), a new alternative is developed for solving linear quadratic (LQ) optimal control problem of discrete-time systems. In this method, the LQ performance index is transformed into the energy function of CHNN, and the control sequence into the output vector of the neurons of CHNN. As a result, solving LQ dynamic optimization problem is equivalent to operating associated CHNN from its initial state to the terminal state. The stable output vector of CHNN represents the optimal control sequence. Because CHNN works in parallel and is of real-time characteristic, the present method is easier to satisfy the requirement of real-time control and will be promising in application.

Finite-horizon and infinite-horizon linear quadratic optimal control problems: A data-driven Euler scheme

Existence, uniqueness, and stability of solutions to singular linear quadratic optimal control problems