Integration Of Riccati Equation Research Articles

Solvability of an Operator Riccati Integral Equation in a Reflexive Banach Space

We show that if X is a reflexive Banach space, then a nonautonomous operator Riccati integral equation has a unique strongly continuous self-adjoint nonnegative solution P(t) in L(X,X^*).

Optimal Sensor Placement: A Robust Approach

We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. This paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem, and finalizes with a range of numerical tests.

Solutions and Approximations to the Riccati Integral Equation with Values in a Space of Compact Operators

In this paper, we present conditions that ensure the existence of Bochner integrable solutions of infinite dimensional Riccati integral equations. In particular, we focus on $\mathscr{I}_p$-valued continuous solutions. We formulate an optimal sensor location problem that is based on optimal filtering and show that when the underlying system is of convection-diffusion type the Riccati integral equation has Bochner integrable solutions. We use these results to approximate the sensor placement problem by using a simple quadrature rule. A numerical example is given to illustrate the results.

The Infinite-Dimensional Optimal Filtering Problem with Mobile and Stationary Sensor Networks

In this article, we introduce a framework to address filtering and smoothing with mobile sensor networks for distributed parameter systems. The main problem is formulated as the minimization of a functional involving the trace of the solution of a Riccati integral equation with constraints given by the trajectory of the sensor network. We prove existence and develop approximation of the solution to the Riccati equation in certain trace-class spaces. We also consider the corresponding optimization problem. Finally, we employ a Galerkin approximation scheme and implement a descent algorithm to compute optimal trajectories of the sensor network. Numerical examples are given for both stationary and moving sensor networks.

On Stability and LQ Control of MJLS With a Markov Chain With General State Space

We study the mean square stability and the LQ control of discrete time Markov Jump Linear Systems where the Markov chain has a general state space. The mean square stability is characterized by the spectral radius of an operator describing the evolution of the second moment of the state vector. Two equivalent tests for the mean square stability are obtained based on the existence of a positive definite solution to a Lyapunov equation and a uniformity result respectively. An algorithm for testing the mean square stability is also developed based on the uniformity result. The finite and infinite horizon LQ problems are considered and their solutions are characterized by appropriate Riccati integral equations. An application to Networked Control Systems (NCS) is finally presented and a simple example is studied via simulation.

A linear quadratic problem with unbounded control in Hilbert spaces

This paper considers optimal control problems for an abstract linear system with a quadratic cost functional in a Hilbert space on a fixed time interval $[0,T],$ $ 0 < T < +\infty$. The controls are unbounded and the state weight operators are indefinite and not necessarily smoothing. Some new inequalities are established and applied to prove that the optimal control and the optimal trajectory are continuous. The equivalences among the solvability of the LQ problems, the two-point boundary value problem and the Fredholm integral equation are proved and the state feedback representation for the optimal control is also given in terms of the solution to the Fredholm integral equation. Finally, we derive the closed-loop synthesis of the optimal control via the solution to the Riccati integral equation which exists under some mild conditions. An explanatory example is given at the end. The results are complementary to those of [15, 1}] and [17].

Linear Quadratic Optimal Control of Time-Varying Systems with Indefinite Costs on Hilbert Spaces

In this paper we consider time-varying linear systems on Hilbert spaces and study the linear quadratic optimal control problem with an indefinite performance criterion over an infinite time interval. An example shows that in contrast to the finite-dimensional situation, in general, the solvability of the optimal control problem does not imply the solvability of the associated integral Riccati equation. Using an operator theoretic approach towards the time-varying integral Riccati equation, we derive equivalent conditions for the solvability of both problems.

Linear quadratic problem with unbounded control in Hilbert spaces

A finite horizon linear quadratic optimal control problem in Hilbert space is considered, in which the controls are unbounded and the state weight operators are allowed to be indefinite and not pecessarily smoothing. It is proved that any optimal control and optimal trajectory are continuous. The closed-loop synthesis for optimal control is given via the solution to Fredholm integral equation and also the solution to Riccati integral equation, which exists under some mild conditions.

ON INFINITE DIMENSIONAL RICCATI INTEGRAL EQUATIONS WITH ALL COEFFICIENTS INDEFINITE

Infinite dimensional Riccati integral equations with all coefficients indefinite are studied in this paper. Relationship between this sort of Riccati integral equations and Fredholm integral equations is established.

Optimal feedback control for a linear-quadratic control problem with a state inequality constraint

In this paper, we consider the linear-quadratic control problem (LQCP) for systems defined by evolution operators with an inequality constraint on the state. It is shown that, under suitable assumptions, the optimal control exists, is unique, and has a closedloop structure. The synthesis of the feedback control requires one to solve two Riccati integral equations.

Initial Boundary Value Problems and Optimal Control for Nonautonomous Parabolic Systems

A large class of linear nonautonomous parabolic systems in bounded domains is considered, with control acting on the boundary through Dirichlet or Neumann conditions, from the point of view of semigroup theory. The results from [Rend. Sem. Mat. Univ. Padova, 78 (1987), pp. 47–107], [On fundamental solutions for abstract parabolic equations, Lecture Notes in Math., Vol. 1223, Springer-Verlag, Berlin, Heidelberg, 1986, pp. 1–11] on abstract homogeneous parabolic Cauchy problems allow operators with varying domains and Hölder continuous coefficients to be handled. A representation formula for solutions corresponding to square integrable control functions is derived and used to solve a linear-quadratic regulator problem over finite time horizon, by a direct study of the associated integral Riccati equation.

Quadratic Control for Stochastic Systems Defined by Evolution Operators and Square Integrable Martingales

The infinite dimensional version of the linear quadratic cost control problem is studied by Curtain and Pritchard [2], Gibson [5] by using Riccati integral equations, instead of differential equations. In the present paper the corresponding stochastic case over a finite horizon is considered. The stochastic perturbations are given by Hilbert valued square integrable martingales and it is shown that the deterministic optimal feedback control is also optimal in the stochastic case. Sufficient conditions are given for the convergence of approximate solutions of optimal control problems.

Feedback control in LQCP with a terminal inequality constraint

This paper considers the linear-quadratic control problem (LQCP) for systems defined by evolution operators with a terminal state inequality constraint. It is shown that, under suitable assumptions, the optimal control exists, is unique, and has a closed-loop structure. The synthesis of the feedback control requires one to solve the integral Riccati equation for the unconstrainted LQCP and a linear integral equation whose solution depends on a real parameter satisfying an additional condition.

The LQCP With a Terminal Inequality Constraint and Unbounded Input and Output Operators

This paper deals with the linear quadratic control problem (LQCP) for infinite dimensional systems with a terminal inequality constraint and unbounded input and output operators. It is shown that under a suitable assumption the optimal control exists, is unique and has feedback form. The synthesis of the feedback requires one to solve the integral Riccati equation for the LQCP without state constraints and a linear integral equation whose solution depends on a real parameter satisfying an additional condition.

Riccati integral equations and non-oscillation of self-adjoint linear systems

Riccati integral equations and non-oscillation of self-adjoint linear systems

Special factorization and Riccati integral equations

A class of operator Riccati integral equations is associated with a factorization problem in a certain Banach algebra. Recent results concerning factorization in this algebra are used to obtain existence, uniqueness, and continuous dependence results for the Riccati equations.

Indefinite Quadratic Cost Optimal Control for Distributed System with Unbounded Control Operator

Based on extensions of linear operator semigroup on Hilbert spaces, the indefinite quadratic optimal control of linear distributed system with unbounded control operator is considered. Necessary and sufficient conditions for wellposedness and solvability are obtained. Closed-loop synthesis of optimal control is proved to be a linear state feedback. The feedback operator function satisfies an integral Riccati equation and has some new properties.

Approximation Schemes for the Linear-Quadratic Optimal Control Problem Associated with Delay Equations

The linear regulator problem for delay equations is discussed. We propose a (theoretical) solution involving Riccati integral equations and then axiomatically discuss a general approximation scheme. The details are given for spline and averaging approximations.

On certain Riccati integral equations and second-order linear oscillation

are closely related is well known. Many important results in the oscillation theory of (1.1) are in fact established by studying either (1.2) or (1.3). Recent studies suggest that a great deal can still be said about (1.2) and (l-3), thus leading to new results on (1.1). See [7-91. Particularly useful in these studies is the theory of differential and integral inequalities. The present work further supports this view point. In this paper, we first give a comparison theorem (Section 2, Theorem 1) for Riccati integral equations of the form (1.3) involving monotonic rearrangements of the function Q(t) = ji q(s) ds. As a corollary, we establish an extension of the well-known oscillation criteria of Hille’s [6] to more general potentials (Section 3, Theorem 4). In Section 4 we elaborate on a method of Hartman to derive a comparison-type result involving the square of the integral j-h q(s) ds and then deduce a Lyapunov-type inequality for disfocality. The preliminaries will be introduced in the appropriate sections. 315 0022-247X/82/020315-16$02.00/O

The Riccati Integral Equations for Optimal Control Problems on Hilbert Spaces

The two Riccati integral equations for linear-quadratic control problems involving evolution operators on Hilbert spaces are derived and shown to have a common solution, which yields the closed-loop structure of the optimal control. Riccati integral equations, instead of differential equations, arise because evolution operators are used to represent system dynamics. The operator representing the closed-loop control perturbs the evolution operator representing the uncontrolled system to produce a second evolution operator, representing the optimally controlled system, hence the two Riccati integral equations in terms of these two evolution operators, respectively.Having both Riccati integral equations facilitates the extension of the analysis of optimal control on finite time intervals to the analysis of optimal control on infinite time intervals, and then existence, uniqueness, and stability results for periodic solutions of the Riccati equations are obtained. Finally, sufficient conditions are given for convergence of approximate solutions of optimal control problems on both finite and infinite time intervals.