This paper is concerned with minimal factorizations of rational matrix functions. The treatment is based on a new geometrical principle. In fact, it is shown that there is a one-to-one correspondence between minimal factorizations on the one hand and certain projections on the other. Considerable attention is given to the problem of stability of a minimal factorization. Also the numerical aspects are discussed. Along the way, a stability theorem for solutions of the matrix Riccati equation is obtained.
Let $X,Y^2 , \cdots ,Y^m$ be analytic vector fields on an analytic n-manifold and $\mathcal{D}$ denote the control system $\dot x(t) = X(x) + \sum_{i = 2}^m {u_i (t)} Y^i (x)$, $x(0) = p$. Our major goal is to give high order, computable, sufficient conditions to assure that the reference solution of $\mathcal{D}$ corresponding to $u \equiv 0$ has value at time t on the boundary of the set of all points attainable at time t for small values $t > 0$. This provides high order sufficient conditions for time optimality when the reference solution is singular. These conditions are phased in terms of elements of the Lie algebra generated by $X,Y^2 , \cdots ,Y^m$. We also show that quite general nonlinear systems can be approximated by systems of the above form and this approximation retains more information than the standard linearization about the reference solution.
The major theorem of this paper is very closely parallel to the classical Pontryagin maximum principle. The classical case, very roughly stated, says that if $u(t)$ is a control function which has an associated trajectory $x(t)$, then there is a function $H(v,x,t)$ such that $u(t)$ is optimal only if for each t and for all v in the control set, \[H(u(t),x(t),t) \leqq H(v,x(t),t).\] Our stochastic case of the open loop problem, stated even more roughly, says that there is a function $H(v,x,t,\omega )$ such that a control function $u(t)$ with associated trajectory $x(t,\omega )$ is optimal only if for all t and for all v in the control set, \[E\{ H(u(t),x(t,\omega ),t,\omega )\} \leqq E\{ H(v,x(t,\omega ),t,\omega )\} .\] Using this result, we then proceed to define a process whereby a control can be tested for optimality in the closed loop case, where information is acquired at a finite number of times. Throughout the paper, the trajectories are determined by a stochastic integral equation. The stochastic integrals used are McShane’s first and second order belated integrals.
We consider stochastic optimal control problems of Mayer type with dynamics in the form of a system of ordinary differential equations perturbed by a countable state Markov process, and we prove the existence of an optimal control in the class of non-anticipative functions. The proof takes the same approach as the "direct" method of the calculus of variations, used extensively in deterministic problems, but substitutes probabilistic concepts where necessary.
This paper develops a theoretical foundation for the numerical solution of two classes of infinite zero sum games, namely continuous and $L^\infty $ games. Our approach is to introduce a dynamical model (a functional differential equation) for nondynamic $L^\infty $ games and then to show that approximate solutions to a symmetric $L^\infty $ game can be obtained by examining the limiting behavior of the game’s dynamical model. By viewing a continuous game as an $L^\infty $ game, it is shown that exact solutions to a symmetric continuous game can be found by examining the limiting behavior of the corresponding $L^\infty $ game dynamical model. Since the dynamical model is nonlinear, a proof of the existence and uniqueness of its solutions is included. Finally a symmetrization is described for continuous and $L^\infty $ games, and thus the theory provides a general method for solving games of these classes.
The problem, to which extent does the closed-loop time-optimal control of a linear system fulfill its task if the system is subject to small perturbations, is studied.
On a rectangular parallelopiped in $R^N $, $N \geqq 2$, we consider the equation $u_{tt} = \Delta u + f(u,u_t )$, where f is a nonlinear perturbation meeting certain conditions. We prove that the above system is locally controllable at $u = 0$, $u_t = 0$; i.e., the set of states in a certain function space which can be reached from $(0,0)$ in a finite time $T < \infty $ using boundary controls is an open neighborhood of $(0,0)$ in that function space. These results generalize to the nonlinear case conclusions obtained by Russell for the linear wave equation, in which global controllability was established.
The Lagrange dual of control problems with linear dynamics, convex cost and convex inequality state and control constraints is analyzed. If an interior point assumption is satisfied, then the existence of a solution to the dual problem is proved; if there exists a solution to the primal problem, then a complementary slackness condition is satisfied. A necessary and sufficient condition for feasible solutions in the primal and dual problems to be optimal is also given. The dual variables p and v corresponding to the system dynamics and state constraints are proved to be of bounded variation while the multiplier corresponding to the control constraints is proved to lie in $\mathcal{L}^1 $. Finally, a control and state minimum principle is proved. If the cost function is differentiable and the state constraints have two derivatives, then the state minimum principle implies that a linear combination of p and v satisfy the conventional adjoint condition for state constrained control problems.
This research is concerned with the asymptotic properties of feedback systems containing random parameters and subjected to stochastic perturbations. For the special class of feedback systems formed by the open loop cascade of a multiplicative white noise, a sector nonlinearity and a convolution operator, conditions are given to insure the stability in the mean square sense of the feedback system. These conditions are expressed in terms of the Fourier transform of the convolution kernel, the sector parameters of the nonlinearity, and the mean and variance parameters of the noise. Their form is reminiscent of the familiar Nyquist criterion and the circle theorem for deterministic systems. The approach adopted is functional analytic in flavor and avoids the use of Markov semigroup techniques and auxiliary Lyapunov functionals.
This paper examines the relationship between the structure of the reachable set for nonlinear systems and the properties of the Lie algebras of vector fields associated with nonlinear systems. An expression for the reachable set at time t is obtained for a large class of nonlinear systems using unbounded controls.
The asymptotic solution to the linear state regulator problem with cheap control is obtained for the situation where the limiting solution is a singular arc of first order and the initial impulse is that of a delta function. The presentation simplifies previous studies and allows generalization to further cases.
We present new necessary conditions for a relaxed minimum in optimal control problems defined by certain classes of ordinary differential equations. These necessary conditions may be helpful in computing singular extremal arcs, in determining when these arcs are “strictly relaxed”, and in defining regions of the state space that contain nonsingular extremal arcs only.
Generalizations of the familiar rank conditions for controllability and observability of linear autonomous finite-dimensional systems to the general case when both the state space and the control space are infinite-dimensional Banach spaces and the operator A acting on the state is only assumed to generate a strongly continuous semigroup (group) are sought. It is shown that a suitable version of the rank condition, although generally only sufficient for approximate controllability (observability), is however "essentially" necessary and sufficient in two important cases: (i) when A generates an analytic semigroup, (ii) when A generates a group. Such generalization of the rank condition is then used to derive, in turn, easy-to-check tests for approximate controllability (observability) for the important class of normal operators with compact resolvent. In the case of finite number of scalar controls (observations), the tests are expressed by a sequence of rank conditions, using the complete set of eigenvectors of A; moreover, they imply that the minimal number of scalar controls (scalar observations) be not less than the highest multiplicity of the eigenvalues of A. Applications to heat equation as well as wave equation types of systems in finite spatial domains are included. The case when A fails to have a compact resolvent is also analyzed in two examples describing the heat equation in infinite spatial domains, by employing a general procedure.
In the paper a general model of a nonlinear network is constructed. The model considered is a generalization of the Hilbert network introduced in [1]. It is assumed that the generalized Hilbert network consists of at most countably many lumped elements described by nonlinear multivalued operators from a subset of a Hilbert space $\mathcal{H}$ into $\mathcal{H}$. Several theorems are proved on the existence and uniqueness of the solution of the network. Also, conditions are established under which the admittance operator of a generalized Hilbert network is causal.
We consider the problem of minimizing a functional of the type \[l(x(0),x(1)) + \int_0^1 {L(t,x,\dot x)dt,} \] where l and L are permitted to attain the value $ + \infty $. We show that many standard variational and optimal control problems may be expressed in this form. In terms of certain generalized gradients, we obtain necessary conditions satisfied by solutions to the problem, in the form of a generalized Euler–Lagrange equation. We also extend the necessary condition of Weierstrass to this setting. The results obtained allow one to treat not only the standard problems but others as well, bringing under one roof the classical (differentiable) situation, the cases where convexity assumptions replace differentiability, and new problems where neither intervene. We apply the results in the final section to derive a new version of the maximum principle of optimal control theory.
We consider linear quadratic games in a Hilbert space. The system equation is linear and involves an unbounded operator which generates a strongly continuous evolution operator (or semigroup). We show that the existence of a solution to a Riccati integral equation implies the existence of a saddle point for the closed-loop game, and that the former is guaranteed if there exists a unique open-loop saddle point. We also consider quadratic games on an infinite interval.
The identification problem of concern here is the estimation of a real function $f(x)$ by means of noisy observations $\{ (X_i ,f(X_i ) + \eta _i (X_i ))\} $ of its pairs, the $X_i $’s being chosen independently according to some fixed law P. The approach taken for estimation is the “potential function” method (its sources are referenced herein), to wit: Choose $f_0 $ arbitrarily and define the sequence $\{ f_n \} $ by the recursive relation $f_{n + 1} (x) = f_n (x) + \gamma _n (f(X_{n + 1} ) + \eta (X_{n + 1} ) - f_n (X_{n + 1} ))K(X_{n + 1} ,x)$, K being a positive symmetric kernel. From earlier publications it is known that under certain mild restrictions $E[\| {f_n - f} \|^2 ] \to 0$ in the $L_2 (p)$-norm. Rates of convergence have been obtained in the restrictive case that $K(x,y) = \sum_{i - 1}^N {\lambda _i^2 } \phi _i (x)\phi _i (y)$ and $f(x) \in {\operatorname{span}}\{ \phi _i ,1 \leqq i \leqq N\} $. The contribution of this paper is to prove that while no uniform bounds exist in the $L_2 (p)$-norm (we prove this) if $\{ \phi _i \} $ is an infinite set, we do have $E[\| {f - f_n } \|_k ^2 ] < C_n (\| f \|)$ for the norm $\| g \|_k^2 = \int {\int g (x)g(y)K(x,y)p(x)p(y)dxdy} $ and $\{ C_n (r)\} $ a sequence converging to 0 for each positive r. A final result concerns the rate at which increasing finite-dimensional projections of $f_n - f$ converge to 0 in the $L_2 (p)$-norm. From our methods it is seen that if $f \notin V = {\operatorname{span}}(\{ \phi _i \} )$, then $f_n $ converges in the mean to the projection of f on V.
The convergence properties for the solution of the discrete time Riccati matrix equation are extended to Riccati operator equations such as arise in a gyroscope noise filtering problem. Stabilizability and detectability are shown to be necessary and sufficient conditions for the existence of a positive semidefinite solution to the algebraic Riccati equation which has the following properties (i) it is the unique positive semidefinite solution to the algebraic Riccati equation, (ii) it is converged to geometrically in the operator norm by the solution to the discrete Riccati equation from any positive semidefinite initial condition, (iii) the associated closed loop system converges uniformly geometrically to zero and solves the regulator problem, and (iv) the steady state Kalman–Bucy filter associated with the solution to the algebraic Riccati equation is uniformly asymptotically stable in the large. These stability results are then generalized to time-varying problems; also it is shown that even in infinite dimensions, controllability implies stabilizability.
Certain measure-theoretic issues raise the possibility that (a) the optimal value of a multistage stochastic programming problem may be ill-defined and (b) the recursion defining the problem may fail, so the problem itself is not even defined. The first difficulty is illustrated by example. A rigorous definition of multistage stochastic programming with fixed linear recourse is shown to avoid this difficulty. In the context of the new definition, certain measurability, convexity, and lower-semicontinuity assumptions on the objective function preclude the second possibility.
Many stochastic control (or parametrized) systems have (expected value) objective functions of largely unknown form, but where noise corrupted observations can be taken at any selected value of a finite-dimensional parameter x . The parameter x must satisfy equality and inequality constraints. The usual numerical techniques of nonlinear programming on control theory are not usually helpful here. The paper discusses a number of algorithms (with convergence proofs) for selecting a sequence of parameter values $\{ X_n \} $, where $X_n $ depends on $X_{n - 1} $ and observations taken at $X_{n - 1} $, and the limit points are both feasible and satisfy the Kuhn–Tucker necessary condition (w.p. 1 (with probability 1)). The algorithms are stochastic “small step” versions of the deterministic combined penalty function-multiplier methods.
