Farzad Rezaei

Optimization of stochastic uncertain systems with variational norm constraints

This paper considers optimization of stochastic uncertain systems on general abstract spaces, when the uncertainty of the system is described by a variational norm constraint. The pay-off is defined as a linear functional of the uncertain measure. By invoking the Hanh-Banach theorem, the maximization problem of the linear functional over the constraint set is shown to correspond to a convex combination of L1 and Linfin norms. Further, the maximizing measure is constructed using a tilted exponential probability measure. The abstract results are subsequently employed to formulate a new class of uncertain continuous-time nonlinear stochastic controlled systems, in which the control seeks to minimize a linear functional while the measure seeks to maximize it over the variational norm constraint set. The variational norm uncertainty model admits uncertainties in both drift and diffusion coefficients of the stochastic differential equation describing the system. Hence it is much more general than existing uncertainty models described by relative entropy constraints.

Nonlinear Estimation for a Class of Systems

This paper considers nonlinear estimation problems for classes of models, and employs relative entropy to describe the uncertainty classes. Two optimization problems are formulated on general Banach spaces, and their solutions are sought: 1) when the transition probability between the signal to be estimated X and the measurement Y or stochastic kernel is unknown, and 2) when the joint probability induced by the random variables (RVs) X, Y is unknown. For both problems, the uncertainty is described by a relative entropy constraint between the unknown distribution and a fixed nominal distribution. The results include existence of the optimal measures using weak convergence techniques, and properties associated with the estimate of the true distribution. Classical examples are chosen to illustrate the applicability of the results.

Stochastic optimal control of discrete-time systems subject to conditional distribution uncertainty

The aim of this paper is to address optimality of control strategies for stochastic discrete time control systems subject to conditional distribution uncertainty. This type of uncertainty is motivated from the fact that the value function involves expectation with respect to the conditional distribution. The issues which will be discussed are the following. 1) Optimal stochastic control systems subject to conditional distribution uncertainty, 2) optimality criteria for stochastic control systems with conditional distribution uncertainty, including principle of optimality and dynamic programming.

Lossless coding with generalized criteria

This paper presents prefix codes which minimize various criteria constructed as a convex combination of maximum codeword length and average codeword length, or, a convex combination of the average of an exponential function of the codeword length and the average codeword length. This framework encompasses as a special case several criteria previously investigated in the literature, while relations to universal coding is discussed. The coding algorithm derived is parametric resulting in re-adjusting the initial source probabilities via a weighted probability vector according to a merging rule. An algorithm is presented to compute the weighting vector.

Stochastic Optimal Control Subject to Ambiguity

The aim of this paper is to address optimality of control strategies for stochastic control systems subject to uncertainty and ambiguity. Uncertainty corresponds to the case when the true dynamics and the nominal dynamics are dierent but they are dened on the same state space. Ambiguity corresponds to the case when the true dynamics are dened on a higher dimensional state space than the nominal dynamics. The paper is motivated by a brief summary of existing methods dealing with optimality of stochastic systems subject to uncertainty, and a discussion on its shortcoming when stochastic systems are ambiguous. The issues which will be discussed are the following. 1) Modeling methods for ambiguous stochastic systems, 2) formulation of optimal stochastic control systems subject to ambiguity, 3) optimality criteria for ambiguous stochastic control systems.

Minimax rate distortion for a class of sources

This paper deals with rate distortion or source coding with fidelity criterion, in measure spaces, for a class of source distributions. The class of source distributions is described by a relative entropy constraint set between the true and a nominal distribution. The rate distortion problem for the class is thus formulated and solved using minimax strategies, which result in robust source coding with fidelity criterion.

Optimal stochastic control of discrete-time systems subject to total variation distance uncertainty

This paper presents another application of the results in [1], [2], where existence of the maximizing measure over the total variation distance constraint is established, while the maximizing pay-off is shown to be equivalent to an optimization of a pay-off which is a linear combination of L1 and L∞ norms. Here emphasis is geared towards to uncertain discrete-time controlled stochastic dynamical system, in which the control seeks to minimize the pay-off while the measure seeks to maximize it over a class of measures described by a ball with respect to the total variation distance centered at a nominal measure. Two types of uncertain classes are considered; an uncertainty on the joint distribution, an uncertainty on the conditional distribution. The solution of the minimax problem is investigated via dynamic programming.

Stochastic optimal control subject to variational norm uncertainty: viscosity subsolution for generalized HJB inequality

This paper is concerned with optimization of stochastic uncertain systems, when systems are described by measures and the pay-off by a linear functional on the space of measure, on general abstract spaces. Robustness is formulated as a minimax game, in which the control seeks to minimize the pay-off over the admissible controls while the measure aims at maximizing the pay-off over the total variational distance uncertainty constraint between the uncertain and nominal measures. This paper is a continuation of the abstract results in [1], where existence of the maximizing measure over the total variational distance constraint is established, while the maximizing payoff is shown to be equivalent to an optimization of a payoff which is a linear combination of L1 and L∞ norms. The maximizing measure is constructed from a convex combination of a sequence of tilted measures and the nominal measure. Here emphasis is geared towards the application of the abstract results to uncertain continuous-time controlled stochastic differential equations, in which the control seeks to minimize the pay-off while the measure seeks to maximize it over the total variational distance constraint. The maximization over the total variational distance constraint is resolved resulting in an equivalent pay-off which is a non-linear functional of the nominal measure of non-standard form. The minimization over the admissible controls of the non-linear functional is addressed by deriving a HJB inequality and viscosity subsolution. Throughout the paper the formulation and conclusions are related to previous work found in the literature.