Min-max game theory and algebraic Riccati equations for boundary control problems with continuous input-solution map. Part II: The general case

We consider the abstract dynamical framework of [LT3, class (H.2)] which models a variety of mixed partial differential equation (PDE) problems in a smooth bounded domain ⁿ , arbitraryn, with boundaryL ₂-control functions. We then set and solve a min-max game theory problem in terms of an algebraic Riccati operator, to express the optimal quantities in pointwise feedback form. The theory obtained is sharp. It requires the usual Finite Cost Condition and Detectability Condition, the first for existence of the Riccati operator, the second for its uniqueness and for exponential decay of the optimal trajectory. It produces an intrinsically defined sharp value of the parameter, here called _c (critical), _c0, such that a complete theory is available for > _c, while the maximization problem does not have a finite solution if 0 < < _c. Mixed PDE problems, all on arbitrary dimensions, except where noted, where all the assumptions are satisfied, and to which, therefore, the theory is automatically applicable include: second-order hyperbolic equations with Dirichlet control, as well as with Neumann control, the latter in the one-dimensional case; Euler-Bernoulli and Kirchhoff equations under a variety of boundary controls involving boundary operators of order zero, one, and two; Schroedinger equations with Dirichlet control; first-order hyperbolic systems, etc., all on explicitly defined (optimal) spaces [LT3, Section 7]. Solution of the min-max problem implies solution of theH -robust stabilization problem with partial observation.The research of C. McMillan was partially supported by an IBM Graduate Student Fellowship and that of R. Triggiani was partially supported by the National Science Foundation under Grant NSF-DMS-8902811-01 and by the Air Force Office of Scientific Research under Grant AFOSR-87-0321.     

Oscillations of nonlinear hyperbolic equations with functional arguments via Riccati method

This paper is devoted to the study of the oscillatory behavior of solutions of nonlinear hyperbolic equations with functional arguments by using integral averaging method and a generalized Riccati technique. First, we establish oscillation results for nonlinear hyperbolic equations by applying oscillation criteria for Riccati inequality. Secondly, we present interval oscillation results for nonlinear hyperbolic equations. These results improve and extend oscillation results of Cui and Xu [1].     

Iterative techniques for Riccati game equations

We present a monotone iterative technique for the computation of a solution of a Riccati-type equation relevant to the theory of differential games. For this purpose, we show that the Kleinman algorithm for Riccati equation computations converges under extremely general conditions.The research reported in this paper was made possible in part through the Division of Engineering and Applied Physics, Harvard University, by the US Office of Naval Research, Joint Electronics Program, Contract No. N00014-75-C-0648, and by the National Science Foundation, Grant No. GK-31511.     

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.     

Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations

Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.     

Input dynamics and nonstandard riccati equations with applications to boundary control of damped wave and plate equations

An optimization problem for a control system governed by an analytic generator with unbounded control actions is considered. The solution to this problem is synthesized in terms of the Riccati operator, arising from a nonstandard Riccati equation. Solvability and uniqueness of the solutions to this Riccati equation are established. This theory is applied to a boundary control problem governed by damped wave and plate equations.Research of this author partially supported by NSF Grant DMS 9204338.     

Almost-periodic solutions for Riccati equations of stochastic control

We consider an average quadratic cost criteria for affine stochastic differential equations with almost-periodic coefficients. Under stabilizability and detectability conditions we show that the Riccati equation associated with the quadratic control problem has a unique almost-periodic solution. In the periodic case the corresponding result is proved in [4].     

Discrete matrix Riccati equations with superposition formulas

An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati equations. Here we shall describe discretizations of Riccati equations that preserve the superposition formulas. The approach is general enough to include q-derivatives and standard discrete derivatives.     

A singular control approach to highly damped second-order abstract equations and applications

In this paper we restudy, by a radically different approach, the optimal quadratic cost problem for an abstract dynamics, which models a special class of second-order partial differential equations subject to high internal damping and acted upon by boundary control. A theory for this problem was recently derived in [LLP] and [T1] (see also [T2]) by a change of variable method and by a direct approach, respectively. Unlike [LLP] and [T1], the approach of the present paper is based on singular control theory, combined with regularity theory of the optimal pair from [T1]. This way, not only do we rederive the basic control-theoretic results of [LLP] and [T1]—the (first) synthesis of the optimal pair, and the (first) nonstandard algebraic Riccati equation for the (unique) Riccati operator which enters into the gain operator of the synthesis—but in addition, this method also yields new results—a second form of the synthesis of the optimal pair, and a second (still nonstandard) algebraic Riccati equation for the (still unique) Riccati operator of the synthesis. These results, which show new pathologies in the solution of the problem, are new even in the finite-dimensional case. This research was made possible by NATO Collaborative Research Grant SA.5-2-05 (CRG.940161) 274/94/JARC-501, whose support is gratefully acknowledged. The research of I. Lasiecka and R. Triggiani was supported also by the National Science Foundation under Grant NSF-DMS-92-04338. The research of L. Pandolfi was written with the programs of GNAFA-CNR. The main results of the present paper were announced in [LPT].     

Formal continued fractions solutions of the generalizedsecond order Riccati equations,applications

Introducing the notion of the formal continued fractions solutions of the generalized second order Riccati equations, one can compute either a rational approximation of the solution or a rational solution and perform a location of the singularities in the complex plane. This revised version was published online in August 2006 with corrections to the Cover Date.     

A note on the regularity of solutions of infinite dimensional Riccati equations

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smooths solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement.     

Backward error,sensitivity, and refinement of computed solutions of algebraic Riccati equations

In this paper, a new backward error criterion, together with a sensitivity measure, is presented for assessing solution accuracy of nonsymmetric and symmetric algebraic Riccati equations (AREs). The usual approach to assessing reliability of computed solutions is to employ standard perturbation and sensitivity results for linear systems and to extend them further to AREs. However, such methods are not altogether appropriate since they do not take account of the underlying structure of these matrix equations. The approach considered here is to first compute the backward error of a computed solution X? that measures the amount by which data must be perturbed so that X? is the exact solution of the perturbed original system. Conventional perturbation theory is used to define structured condition numbers that fully respect the special structure of these matrix equations. The new condition number, together with the backward error of computed solutions, provides accurate estimates for the sensitivity of solutions. Optimal perturbations are then used in an iterative refinement procedure to give further more accurate approximations of actual solutions. The results are derived in their most general setting for nonsymmetric and symmetric AREs. This in turn offers a unifying framework through which it is possible to establish similar results for Sylvester equations, Lyapunov equations, linear systems, and matrix inversions.     

Mild solutions for abstract fractional differential equations

We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a,?k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C ₀-semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behaviour of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley–Torvik equations.     

A modified Newton method for solving non-symmetric algebraic Riccati equations arising in transport theory

Liang Bao ^{The non-symmetric algebraic Riccati equation arising in transporttheory can be rewritten as a vector equation and the minimalpositive solution of the non-symmetric algebraic Riccati equationcan be obtained by solving the vector equation. In this paper,we apply the modified Newton method to solve the vector equation.Some convergence results are presented. Numerical tests showthat the modified Newton method is feasible and effective, andoutperforms the Newton method.}     

On the convergence of inexact Newton methods for discrete-time algebraic Riccati equations

In this paper, we present a convergence analysis of the inexact Newton method for solving Discrete-time algebraic Riccati equations (DAREs) for large and sparse systems. The inexact Newton method requires, at each iteration, the solution of a symmetric Stein matrix equation. These linear matrix equations are solved approximatively by the alternating directions implicit (ADI) or Smith?s methods. We give some new matrix identities that will allow us to derive new theoretical convergence results for the obtained inexact Newton sequences. We show that under some necessary conditions the approximate solutions satisfy some desired properties such as the d-stability. The theoretical results developed in this paper are an extension to the discrete case of the analysis performed by Feitzinger et al. (2009) [8] for the continuous-time algebraic Riccati equations. In the last section, we give some numerical experiments.     

Adapted Riccati technique and non-oscillation of linear and half-linear equations

The aim of this paper is to mention a generalization of the adapted Riccati equation and, using this method, to prove a non-oscillatory result concerning half-linear differential equations with coefficients having mean values. Note that this result is new even for linear equations.     

Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

In this paper, we consider large‐scale nonsymmetric differential matrix Riccati equations with low‐rank right‐hand sides. These matrix equations appear in many applications such as control theory, transport theory, applied probability, and others. We show how to apply Krylov‐type methods such as the extended block Arnoldi algorithm to get low‐rank approximate solutions. The initial problem is projected onto small subspaces to get low dimensional nonsymmetric differential equations that are solved using the exponential approximation or via other integration schemes such as backward differentiation formula (BDF) or Rosenbrock method. We also show how these techniques can be easily used to solve some problems from the well‐known transport equation. Some numerical examples are given to illustrate the application of the proposed methods to large‐scale problems.