Game-theoretic coupled riccati equations associated to controlled linear differential systems with jump markov perturbations

In this paper we investigate several properties of the stabilizing solution of a class of systems of Riccati type differential equations with indefinite sign associated to controlled systems described by differential equations with Markovian jumping.

We show that the existence of a bounded on R ₊ and stabilizing solution for this class of systems of Riccati type differential equations is equivalent to the solvability of a control-theoretic problem, namely disturbance attenuation problem.

If the coefficients of the considered system are theta;-periodic functions then the stabilizing solution is also theta;-periodic and if the coefficients are asymptotic almost periodic functions, then the stabilizing solution is also asymptotic almost periodic and its almost periodic component is a stabilizing solution for a system of Riccati type differential equations defined on the whole real axis. One proves also that the existence of a stabilizing and bounded on R ₊ solution of a system of Riccati differential equations with indefinite sign is equivalent to the existence of a solution to a corresponding system of matrix inequalities. Finally, a minimality property of the stabilizing solution is derived.     

A class of discrete time generalized Riccati equations

In this paper, a class of discrete-time backward non-linear equations defined on some ordered Hilbert spaces of symmetric matrices is considered. The problem of the existence of some global solutions is investigated. The class of considered discrete-time non-linear equations contains, as special cases, a great number of difference Riccati equations both from the deterministic and the stochastic framework. The results proved in the paper provide the sets of necessary and sufficient conditions that guarantee the existence of some special solutions of the considered equations as: the maximal solution, the stabilizing solution and the minimal positive semi-definite solution. These conditions are expressed in terms of the feasibility of some suitable systems of linear matrix inequalities (LMI). One shows that in the case of the equations with periodic coefficients to verify the conditions that guarantee the existence of the maximal or the stabilizing solution, we have to check the solvability of some systems of LMI with a finite number of inequations. The proofs are based on some suitable properties of discrete-time linear equations defined by the positive operators on some ordered Hilbert spaces chosen adequately. The results derived in this paper provide useful conditions that guarantee the existence of the maximal solution or the stabilizing solution for different classes of difference matrix Riccati equations involved in many problems of robust control both in the deterministic and the stochastic framework. The proofs are deterministic and are accessible to the readers less familiarized with the stochastic reasonings.     

Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping

In this paper, a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled Riccati differential equations arising in connection with control of some linear stochastic systems is considered.The system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite-dimensional Hilbert space adequately choosen.This allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of Lyapunov type.Our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions.Conditions providing the existence of a maximal solution (minimal solution respectively) with respect to some classes of global solutions are presented. It is shown that if the coefficients of the equations are periodic functions all these special solutions (stabilizing, maximal, minimal) are periodic functions, too.Whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves.     

Computation of the stabilizing solution of game theoretic Riccati equation arising in stochastic <Emphasis Type="Italic">H</Emphasis><Subscript> ∞ </Subscript> control problems

In this paper, the problem of the numerical computation of the stabilizing solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H _∞ control problem for a class of stochastic systems affected by state dependent and control dependent white noise. The stabilizing solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilizing solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.     

Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability

In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see [2], for finite dimensional stochastic equations or [21], for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see [10], [18]). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on ℝ ₊ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known [18] that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see [10]).      

Exponential Stability for Discrete Time Linear Equations Defined by Positive Operators

In this paper the problem of exponential stability of the zero state equilibrium of a discrete-time time-varying linear equation described by a sequence of linear positive operators acting on an ordered finite dimensional Hilbert space is investigated. The class of linear equations considered in this paper contains as particular cases linear equations described by Lyapunov operators or symmetric Stein operators as well as nonsymmetric Stein operators. Such equations occur in connection with the problem of mean square exponential stability for a class of difference stochastic equations affected by independent random perturbations and Markovian jumping as well us in connection with some iterative procedures which allow us to compute global solutions of discrete time generalized symmetric or nonsymmetric Riccati equations. The exponential stability is characterized in terms of the existence of some globally defined and bounded solutions of some suitable backward affine equations (inequalities) or forward affine equations (inequalities).     

Stochastic partial differential equations in bounded domains with Dirichlet boundary conditions

In this paper we prove the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions. The main tool is an equivalence result, provided by the stochastic characteristics method, between the stochastic equations under investigation and a class of deterministic parabolic equations with moving boundaries, depending on random coefficients. We show the existence of the solution to this last problem, thus providing a solution to the former.     

Solutions for the Linear-Quadratic Control Problem of Markov Jump Linear Systems

The paper is concerned with recursive methods for obtaining the stabilizing solution of coupled algebraic Riccati equations arising in the linear-quadratic control of Markovian jump linear systems by solving at each iteration uncoupled algebraic Riccati equations. It is shown that the new updates carried out at each iteration represent approximations of the original control problem by control problems with receding horizon, for which some sequences of stopping times define the terminal time. Under this approach, unlike previous results, no initialization conditions are required to guarantee the convergence of the algorithms. The methods can be ordered in terms of number of iterations to reach convergence, and comparisons with existing methods in the current literature are also presented. Also, we extend and generalize current results in the literature for the existence of the mean-square stabilizing solution of coupled algebraic Riccati equations.     

A New Approach to Linearly Perturbed Riccati Equations Arising in Stochastic Control

In this paper a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied. Via the concepts of mean square stabilizability and mean square detectability we improve previous results on both the convergence properties of the linearly perturbed Riccati differential equation and the solutions of the linearly perturbed algebraic Riccati equation. Furthermore, our approach unifies, in some way, the study for this class of Riccati equations with the one for classical theory, by eliminating a certain inconvenient assumption used in previous works (e.g., [10] and [26]). The results are derived under relatively weaker assumptions and include, inter alia, the following: (a) An extension of Theorem 4.1 of [26] to handle systems not necessarily observable. (b) The existence of a strong solution, subject only to the mean square stabilizability assumption. (c) Conditions for the existence and uniqueness of stabilizing solutions for systems not necessarily detectable. (d) Conditions for the existence and uniqueness of mean square stabilizing solutions instead of just stabilizing. (e) Relaxing the assumptions for convergence of the solution of the linearly perturbed Riccati differential equation and deriving new convergence results for systems not necessarily observable. Accepted 30 July 1996     

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

We consider the controlled stochastic Navier–Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier–Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists.     

Backward Stochastic Riccati Equations and Infinite Horizon L-Q Optimal Control with Infinite Dimensional State Space and Random Coefficients

We study the Riccati equation arising in a class of quadratic optimal control problems with infinite dimensional stochastic differential state equation and infinite horizon cost functional. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context backward stochastic Riccati equations are backward stochastic differential equations in the whole positive real axis that involve quadratic non-linearities and take values in a non-Hilbertian space. We prove existence of a minimal non-negative solution and, under additional assumptions, its uniqueness. We show that such a solution allows to perform the synthesis of the optimal control and investigate its attractivity properties. Finally the case where the coefficients are stationary is addressed and an example concerning a controlled wave equation in random media is proposed.     

一类高维Riccati方程的解

研究了一类具有指数型二分性的高维Riccati方程存在有界解、周期解的充分条件,得到一些结论。     

Global adapted solution of one-dimensional backward stochastic Riccati equations,with application to the mean–variance hedging

Backward stochastic Riccati equations are motivated by the solution of general linear quadratic optimal stochastic control problems with random coefficients, and the solution has been open in the general case. One distinguishing difficult feature is that the drift contains a quadratic term of the second unknown variable. In this paper, we obtain the global existence and uniqueness result for a general one-dimensional backward stochastic Riccati equation. This solves the one-dimensional case of Bismut–Peng's problem which was initially proposed by Bismut (Lecture Notes in Math. 649 (1978) 180). We use an approximation technique by constructing a sequence of monotone drifts and then passing to the limit. We make full use of the special structure of the underlying Riccati equation. The singular case is also discussed. Finally, the above results are applied to solve the mean–variance hedging problem with general random market conditions.     

Stochastic H2 / H∞ Control for Poisson Jump-Diffusion Systems

This paper is concerned with stochastic H_2/H_∞ control problem for Poisson jump-diffusion systems with(x, u, v)-dependent noise, which are driven by Brownian motion and Poisson random jumps. A stochastic bounded real lemma(SBRL for short) for Poisson jump-diffusion systems is firstly established, which stands out on its own as a very interesting theoretical problem. Further, sufficient and necessary conditions for the existence of a state feedback H_2/H_∞ control are given based on four coupled matrix Riccati equations. Finally, a discrete approximation algorithm and an example are presented.     

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.     

Existence and comparison theorems for partial differential equations of Riccati type

In this paper, we discuss the partial differential equation of Riccati type that describes the optimal filtering error covariance function for a linear distributed-parameter system with pointwise observations. Since this equation contains the Dirac delta function, it is impossible to apply directly the usual methods of functional analysis to prove existence and uniqueness of a bounded solution. By using properties of the fundamental solution and the classical technique of successive approximation, we prove the existence and uniqueness theorem. We then prove the comparison theorem for partial differential equations of Riccati type. Finally, we consider some applications of these theorems to the distributed-parameter optimal sensor location problem.     

Indefinite stochastic LQ control with cross term via semidefinite programming

An indefinite stochastic linear-quadratic (LQ) optimal control problem with cross term over an infinite time horizon is studied, allowing the weighting matrices to be indefinite. A systematic approach to the problem based on semidefinite programming (SDP) and related duality analysis is developed. Several implication relations among the SDP complementary duality, the existence of the solution to the generalized Riccati equation and the optimality of LQ problem are discussed. Based on these relations, a numerical procedure that provides a thorough treatment of the LQ problem via primal-dual SDP is given: it identifies a stabilizing optimal feedback control or determines the problem has no optimal solution. An example is provided to illustrate the results obtained.     

Mean–variance portfolio selection under a constant elasticity of variance model

This paper discusses a mean–variance portfolio selection problem under a constant elasticity of variance model. A backward stochastic Riccati equation is first considered. Then we relate the solution of the associated stochastic control problem to that of the backward stochastic Riccati equation. Finally, explicit expressions of the optimal portfolio strategy, the value function and the efficient frontier of the mean–variance problem are expressed in terms of the solution of the backward stochastic Riccati equation.     

Optimal control for linear discrete-time systems with Markov perturbations in Hilbert spaces

Email: vio{at}utgjiu.ro Received on September 12, 2007; Accepted on December 26, 2008 In this article, we discuss a quadratic control problem forlinear discrete-time systems with Markov perturbations in Hilbertspaces, which is linked to a discrete-time Riccati equationdefined on certain infinite-dimensional ordered Banach space.We prove that under stabilizability and stochastic uniform observabilityconditions, the Riccati equation has a unique, uniformly positive,bounded on N and stabilizing solution. Based on this result,we solve the proposed optimal control problem. An example illustratesthe theory.     

Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria

In this paper, we consider a linear–quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward–backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which could be different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of an open-loop saddle point coincides with that for the closed-loop saddle point, which leads to the conclusion that the closed-loop representation of an open-loop saddle point is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear–quadratic optimal control problem, the closed-loop representation of an open-loop optimal control coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist.