Hereditary Differential Systems Research Articles

Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift

We establish the existence of smooth densities for solutions of $\mathbf{R}^d$-valued stochastic hereditary differential systems of the form $dx(t) = H(t,x)dt + g(t,x(t - r))dW(t).$ In the above equation, $W$ is an $n$-dimensional Wiener process, $r$ is a positive time delay, $H$ is a nonanticipating functional defined on the space of paths in $\mathbf{R}^d$ and $g$ is an $n \times d$ matrix-valued function defined on $\lbrack 0,\infty) \times \mathbf{R}^d$, such that $gg^\ast$ has degeneracies of polynomial order on a hypersurface in $\mathbf{R}^d$. In the course of proving this result, we establish a very general criterion for the hypoellipticity of a class of degenerate parabolic second-order time-dependent differential operators with space-independent principal part.

Finite-Dimensional Compensators for Infinite-Dimensional Systems via Galerkin-Type Approximation

In this paper existence and construction of stabilizing compensators for linear time-invariant systems defined on Hilbert spaces are discussed. An existence result is established using Galerkin-type approximations in which independent basis elements are used instead of the complete set of eigenvectors. A design procedure based on approximate solutions of the optimal regulator and optimal observer via Galerkin-type approximation is given and the Schumacher approach is used to reduce the dimension of compensators. A detailed discussion for parabolic and hereditary differential systems is included.

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

In this paper operator Riccati equations of evolution and algebraic types arising in the linear quadratic optimal control problem for hereditary differential systems are considered. Regularity results of their solutions are obtained. Especially, for the system with single point delay we show that the optimal feedback kernel function Π 10 is C ∞. This study is motivated by the numerical experiments of K. Ito and R. Teglas (“Legendre-Tau Approximation for Functional Differential Equations. Part II. The Linear Quadratic Optimal Control Problem,” SIAM J. Control and Optimization, 25, 1987 , 1379–1408) in computing the optimal feedback gain via an algebraic Riccati equation, which indicate that such a function is smooth. The regularity result is essential in obtaining a rate of convergence of numerical approximations of the optimal feedback gain operator.

Legendre-Tau Approximation for Functional Differential Equations Part II: The Linear Quadratic Optimal Control Problem

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Chandrasekhar Equations for Infinite Dimensional Systems

In this paper we derive the Chandrasekhar equations for linear time invariant systems defined on Hilbert spaces using a functional analytic technique. An important consequence of this is that the solution to the evolutional Riccati equation is strongly differentiable in time and one can define a “strong” solution of the Riccati differential equation. A detailed discussion on the linear quadratic optimal problem for hereditary differential systems is also included.

Addendum: Stable and Regular Reachability of Relaxed Hereditary Differential Systems

This paper characterizes regular reachability of relaxed hereditary differential systems as a positive controllability property of an associated linear system in the Sobolev space $W^{1,2} $. Thus the results of F. Colonius [SIAM J. Control Optim., 20 (1982), pp. 675–694] are improved. Regular reachability is relevant as a regularity condition in the proof of a maximum principle for fixed final state optimal control problems.

Hereditary realizations of linear neutral delay differential systems

The realization problem for hereditary linear neutral delay differential systems is considered in the context of systems over commutative rings. The notion of formal spectral minimality is introduced and related to the general notion of spectral minimality of infinite dimensional systems. It is shown that for neutral delay systems, unlike retarded delay systems, absolute minimality does not imply formal spectral minimality. For systems with commensurable delays it is shown that an absolutely minimal system may always be reduced to a formally spectrally mininal system by an extended type of system transformation which we call weak restricted system equivalence following the terminology of Rosenbrock. This result is used to give a two-step realization procedure for the construction of canonical neutral realizations for a large class of non-rational transfer functions. A counterexample is given to show that, in the case of non-commensurable delays, there may not exist an hereditary realization which is both absolutely minimal and (formally) spectrally minimal.

Optimal Input/Output Feedback with Structure Constraints

An optimal constrained output feedback problem is studied in the framework of Hilbert resolution space valued random processes. The main result, a necessary condition for optimality, is simultaneously applicable to continuous time, sampled data, finite and infinite dimensional systems. When applied to finite dimensional systems this condition rediscovers in a unified setting most of the classical results already available in the technical literature; when applied to infinite dimensional systems, such as hereditary differential systems, it gives new appropriate extensions of these results.

Linear-Quadratic Optimal Control of Hereditary Differential Systems: Infinite Dimensional Riccati Equations and Numerical Approximations

Recent theory of infinite dimensional Riccati equations is applied to the linear-quadratic optimal control problem for hereditary differential systems, and it is shown that, for most such problems, the operator solutions of the Riccati equations are of trace class (i.e., nuclear). With special attention to trace-norm convergence, an abstract approximation theory is developed and applied to a particular approximation scheme. Numerical examples are given.Problems on both finite and infinite time intervals are studied. For both the hereditary system and the approximating systems in the infinite time problem, characteristic equations are derived for the closed-loop eigenvalues, and formulas for the corresponding eigenvectors are given.

Stable and Regular Reachability for Relaxed Hereditary Differential Systems

We investigate reachability properties of relaxed nonlinear hereditary differential systems in the state space $W^{n,\infty } [ - r,0]$. Reachability of the relaxed system is equivalent to approximate reachability of the ordinary system. Local reachability for the relaxed system can be obtained under much weaker conditions than for the ordinary system. The notion of stable reachability allows us to relate reachability of points in $\mathbb{R}^n $ and states in $W^{n,\infty } [ - r,0]$, Regular reachability is of primary importance in optimal control problems with function space end condition, because in case of regularity Lagrange multipliers in the dual of $W^{n,\infty } [ - r,0]$ can be identified with functions in $W^{n,\infty } [ - r,0]$. Regularly reachable final states and regular trajectories are characterized and it is shown that regularity is a generic property of trajectories. Examples are given where all trajectories reaching a certain final state are regular.

The Maximum Principle for Relaxed Hereditary Differential Systems with Function Space end Condition

This paper contains a proof of the global pointwise maximum principle for relaxed hereditary differential systems with general function space end condition. First a multiplier theorem establishes the existence of Lagrange multipliers $(l_0 ,l)$, where to $l_0 \in \mathbb{R}_ + $ and l is in the dual of the Sobolev space $W^{n,\infty } [ - r,0]$. Then l can be identified with an element of $W^{n,\infty } [ - r,0]$ provided that the optimal trajectory satisfies a certain regularity condition. This yields two equivalent forms of the maximum principle. Using the results on regular reachability obtained in a companion paper, the maximum principle is shown to be—in a certain sense—generically valid.

Optimal Control of Functional Differential Systems

This paper presents a unified approach to diverse optimal control problems for hereditary differential systems (HDS). An abstract local maximum principle is established via the Dubovitskii–Milyutin method. It yields necessary conditions for the optimal control of HDS towards surfaces in $R^n $ and towards target sets in function spaces. Nondegeneracy criteria are included. It is shown that the necessary conditions are sufficient in the case of linear HDS with convex cost functionals. Analogous results are obtained for systems described by Fredholm equations with general control action. For Fredholm systems with targets in function spaces, the attainability space $\mathcal{A}$ is investigated, criteria for $\mathcal{A}$ to be closed are established and full attainability is characterized.

F-reduction of the operator Riccati equation for hereditary differential systems

A number of independent treatments of the linear quadratic optimal control problem for retarded systems are available in the literature. Namely (a) using abstract theory of evolution equations in a Hilbert space, (b) via the theory of Fredholm integral equations, (c) via the Bellman-Hamilton-Jacobi equation. These treatments result in different characterizations of the optimal system via complicated Riccati equations. It is shown that by introducing a certain ‘hereditary operator’ F one can characterize more precisely the structure of the solution of the operator Riccati equation. This in turn provides a missing link between the existing theories, and results in a simplification and in some reduction in the Riccati equation.

State theory of linear hereditary differential systems

In this paper we present a state theory for a class of linear functional differential equations of the retarded type considered by Delfour and Mitter ( J. Differential Equations, 18 1975, 18–28) with initial functions in the product space M p = X × L p (− b, 0; X). Roughly speaking, the state at time t is a piece of trajectory defined over an interval [ t − b, t] for a fixed b > 0. From a study of the properties of the state in M p an operational differential equation, the so-called state equation, is derived in order to describe its evolution. An adjoint state equation is also introduced for the adjoint state and the connection between solutions of the hereditary adjoint system and those of the adjoint state equation is established. All this provides the appropriate framework for the solution and the numerical approximation of the associated linear-quadratic optimal control and filtering problems.

F-Reduction of the Operator Riccati Equations for Hereditary Differential Systems

F-Reduction of the Operator Riccati Equations for Hereditary Differential Systems

Constrained control problems with convex cost in Hilbert space

In recent papers [3, 41 necessary and sufficient conditions were derived for convex control problems involving linear differential equations in Hilbert spaces. In this paper we show that, under relatively weak assumptions (see Section l), one can derive the above mentioned optimality conditions, for a wide class of control problems for linear evolution equations in Hilbert space. The plan of the paper is as follows. Section 1 is concerned principally with the problem formulation. The necessary and sufficient conditions for optimality are in the “subdifferential” form and do not require the differentiability of cost functional. Thus our result is very much in the spirit of Rockafellar works [15, 16-j. These conditions are spelled out in Theorem 1, whose proof is set forth in Section 3. In Section 4 we formulate a control problem for linear hereditary differential systems with convex criterion. The necessary and sufficient conditions are then specialised for this particular problem (see Theorem 2). These results may be compared with those of R. Datko [7,8], H. T. Banks and M. Q. Jacobs [l], H. T. Banks and G. A. Kent [2], M. C. Delfour and S. K. Mitter [9], A. Halanay [l l] (further references may be found in these papers). However, our results on necessary optimality conditions in problems involving linear evolution equations in Hilbert space setting (see Lions’s book [ 121 for significant results in this field) differ from previous results (even specialised to functional differential equations) which involve stronger regularity assumptions than ours. For the most part, these papers are concerned with quadratic cost criterion and certainly the methods used here are very different. Our approach is much similar to that used by the author in [3] and it relies on the methods of convex analysis. For significant results in this field, relevant to this paper we refer to the books of Rockafellar [13] and BrCzis [6].

The linear quadratic optimal control problem for hereditary differential systems: Theory and numerical solution

In the first part of this paper, we summarize and complete earlier results of Delfour-Mitter [3] on the optimal control problem for linear hereditary differential systems (HDS) with a linear-quadratic cost function. The properties of the operator Π(t) which characterizes the feedback gains and the reference functionr(t) are fully detailed. In a second part, we construct an approximation to the linear HDS in state form and to the linear adjoint state equation and prove convergence. In a third part we construct and solve the approximate optimal control problem following the method of J. C. Nedelec. In the last part we construct the approximation to Π(t) andr(t) and prove convergence. Finally we give a number of typical examples to illustrate the main features of the kernel of Π(t).

The Infinite Dimensional Riccati Equation with Applications to Affine Hereditary Differential Systems

The infinite-dimensional versions of the linear quadratic cost control problem and of the linear filtering problem lead to an infinite-dimensional Riccati equation with unbounded operators. Existence and uniqueness theorems for mild solutions of these were established in The infinite dimensional Riccati equations, Ruth F. Curtain and A. J. Pritchard (to appear in J. Math. Anal. Appl.) using a semigroup and evolution operator approach. Although this formulation was very general, covering a large class of parabolic partial differential control systems, it does not cover the semigroup formulation of linear hereditary differential equations introduced by Delfour and Mitter. This paper remedies this and applies the theory to the linear quadratic cost control problem for the afBne linear hereditary differential case.

Hereditary differential systems with constant delays. II. A class of affine systems and the adjoint problem

In this paper we (i) specialize some of the results of Delfour and Mitter ( J. Differential Equations, 12, 1972, 213–235) to a class of representable affine hereditary differential systems, (ii) introduce the hereditary adjoint system, and (iii) give an integral representation of solutions.

Stability and the Infinite-Time Quadratic Cost Problem for Linear Hereditary Differential Systems

This paper studies the infinite-time quadratic cost control problem for a general class of linear autonomous hereditary differential systems. It uses an approach which clarifies the system-theoretic relationship between stabilizability, stability and existence of a solution of an associated operator equation of Riccati type. For this purpose the stability problem is studied and an operator equation of the Lyapunov type is derived. In both cases we obtain equations which characterize the kernels of the Lyapunov and the Riccati equations.