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.
