A. V. Balakrishnan

On stochastic bang bang control

We consider a constrained stochastic control problem (stochastic bang-bang control) studied recently by many authors Ruzicka [1], Zwonkin-Krylov [2], and Ikeda-Watanabe [3], among others. Our techniques are quite different from theirs, and we obtain in addition some results complementing theirs which would appear to be of independent interest as well.

Strong stabilizability and the steady state Riccati equation

We explore strong stabilizability (as opposed to exponential stabilizability) with the aid of the steady state Riccati equation. We show that the latter can have at most one strongly stable solution and obtain some sufficient conditions for existence. We also indicate an application to steady state Kalman filtering where the observation operator is compact so that we may not have exponential stability.

Likelihood ratios for signals in additive white noise

We present a formula for likelihood functionals for signals in which the corrupting noise is modelled as white noise rather than the usual Wiener process. The main difference is the appearance of an additional term corresponding to the conditional mean square error. By way of one application we consider the ‘order-disorder’ problem of Shiryayev.

Radon-Nikodym derivatives of a class of weak distributions on Hilbert spaces

The problem of derivatives of weak distributions is studied in the context of likelihood ratios of signals in noise, the ‘independent’ case. We show that the derivative is defined in that case and obtain a formula for it. The main result is in Section 2; the necessary introductory material is in Section 1. The application to the linear case is given in Section 3, and in Section 4, a non-linear example, in which we show for the first time that the correction term in the white noise version of the Girsanov formula is a random variable whose expected value is the mean square estimation error.

Stochastic optimization theory in Hilbert spaces—1

We study a class of stochastic optimization problems in which the state as well as the observation spaces are permitted to be (Hilbert spaces) of non-finite dimension. Although there have been previous attempts in the Hilbert space setting, our results, techniques, as well as applications, are totally different. We initiate the use of Gauss measure on a Hilbert space even though it is only finitely additive; and an associated theory of white noise, in contrast to the Wiener process theory, which is novel even in the finite dimensional case. We only treat time-invariant systems, but no strong ellipticity or coercivity conditions are used; we exploit the theory of semigroups of operators in contrast to the Lions-Magenes theory. A key result involves a far-reaching generalization of the Factorization theorem of Krein. We apply the results to the problem of boundary observation and control for partial differential equations. By the creation of a special state space, we can apply the theory to problems in which the state equations are finitedimensional but the noise does not have a rational spectrum.In a final section, we present a stochastic theory for inverse problems (System Identification) in the Hilbert space setting. The basic theoretical problem is the calculation of R-N derivatives for finitely additive measures. A fundamental result concerns Identifiability; in particular the identifiability of diffusion coefficients from boundary data is treated here for the first time.

On a generalization of the Kalman-Yakubovic lemma

In this paper we generalize the Kalman-Yakubovic lemma to infinite dimensions—or, more precisely, to semigroups of operators over a Hilbert space. The proof differs substantially from the finite-dimensional version and is based on the Paley-Wiener-Helson-Lowdenslager factorization theorem.

Damping operators in continuum models of flexible structures : explicit models for proportional damping in beam bending with end-bodies

A convenient “working” model for passive damping in a flexible structure is proportional damping. Strictly proportional damping requires that the damping operator be (essentially) the square root of the stiffness operator. In this paper we present an explicit calculation of the square root for the case of the bending of a uniform Bernoulli beam clamped at one end and subject to control forces and moments at the other end, and we show that nonlocal terms are added in the interior as well as at the ends in contrast to the case where there are no end-masses and both ends are simply supported. We show that if strict proportionality is relaxed to require only asymptotic proportionality, then we can avoid the nonlocal feature although the boundary equations will still need to include additional terms.

On abstract stochastic bilinear equations with white noise inputs

We study the response of a class of stochastic bilinear hyperbolic (oscillatory) systems to multiplicative noise inputs. We formulate them as bilinear abstract Cauchy problems in the framework of (finitely additive) white noise theory. We show that the solution can be expressed in terms of the integral of the noise, which enables us to study relevant stochastic properties of the solution.

On a class of Riccati equations in a Hilbert space

We establish existence and uniqueness of solutions of a class of Riccati equations in Hilbert space oćcurring in filtering problems for distributed parameter systems using “point” sensors.

A likelihood ratio formula for random fields with application to physical geodesy

A formula for the likelihood ratio applicable to estimation and inference problems arising in random field data models in physical geodesy is derived based on multiparameter white noise theory.