Yaakov Yavin

On the optimal control of a stochastic system with discontinuous sample paths

This paper deals with the optimal control of a stochastic n-order system. It is assumed that the system is subjected to two different kinds of perturbations. The first kind of perturbation is represented by a vector of independent standard Wiener processes and the second kind by a vector of a generalized type of Poisson process. By applying the calculus of variations necessary conditions on the optimal controls are derived. These conditions are given by a pair of coupled non-linear partial integro-differential equations, A stochastic second-order system is given, as a test case, and a numerical method for the computation of its optimal controls is suggested. The efficiency and applicability of this method ore demonstrated with examples.

On the numerical solution of two coupled nonlinear partial integro-differential equations related to the optimal control of a nonlinear noisy oscillator

Abstract This paper deals with the optimal control of a nonlinear stochastic thrid-order oscillator. It is shown that, in order to implement the optimal feedback control law, a set of two coupled nonlinear partial integro-differential equations has to be solved. A finite difference algorithm for the solution of these equations is proposed, and its efficiency and applicability are demonstrated with examples.

Optimal Controls that Maximize the Expectation of First Passage Time

Abstract This paper deals with optimal controls that maximize the expectation of first passage time of the state, of a stochastic non-linear system, to the complement of an open and bounded domain. The performance index includes a penalty on the magnitude of the deviation of the first passage time from its expectation. The nonlinear system considered here is subjected to two different kinds of perturbations. The first kind of perturbation is represented by a vector of independent standard Wiener processes and the second kind by a vector of a generalized type of Poisson process. By using a variational approach, necessary conditions on the optimal controls are derived. These conditions are given by a set of four coupled nonlinear partial integro- differential equations. A nonlinear stochastic third-order system is given as a test case, and a numerical method for the computation of its optimal controls, is suggested. The efficiency and applicability of this method are demonstrated with examples.

Computation of functionals over discontinuous sample paths of a stochastic van der pol oscillator

Abstract This paper deals with a random van der Pol oscillator. It is assumed that the oscillator is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and the second kind by a homogeneous process with independent increments, finite second order moments, mean zero and no continuous sample functions. In order to measure quantitatively the stochastic stability of the oscillator, two functionals are defined over its phase plane sample paths. It is shown that each of these functionals is a solution to a corresponding partial integro-differential equation. A numerical procedure for the solution of these equations, is suggested, and its efficiency and applicability are demonstrated with examples.

On the numerical solution of a nonlinear partial differential equation related to the optimal control of a noisy oscillator

Abstract This paper deals with the optimal control of a linear oscillator with parametric excitation. It is shown that, in order to implement the optimal feedback control law, a nonlinear partial differential equation has to be solved. A finite difference algorithm for the solution of this equation is proposed, and its efficiency and applicability are demonstrated with examples.

On the modelling and stability of a stochastic distributed parameter system

Abstract A stochastic non-linear distributed parameter system of a parabolic typo is dealt with. The system is stochastic due to a distributed multiplicative gain. The gain is a distributed white (in time) Gaussian noise. The distributed parameter system is modelled by two different typos of stochastic partial differential equations. By using a stochastic Liapunov type functional, sufficient conditions for a weak stability of the systems sample functions, are derived. It is shown that these conditions depend heavily on the type of the stochastic partial differential equation which is selected in order to model the system. The conditions for stochastic stability, for the two different models, are compared. Also, for the sake of comparison, a deterministic version of the problem is dealt with.

On the stochastic stability of a parabolic type system

A stochastic distributed parameter system of a parabolic type is dealt with. The system is stochastic duo to a distributed multiplicative gain. The gain is a non-linear function of a Wiener process. By applying a finite-difference scheme on the spatial coordinate, and using a stochastic Linpunov type functional, sufficient conditions for n weak stability of the systems solutions, are derived.

On the suboptimal control of a heat conduction process

An optimal control problem, concerned with a heat conduction process of a moving material in a furnace, is posed. Due to the stochastic nature of the process only a suboptimal version of the problem is treated. This is done in the following manner. First the process is modelled by using an adequate stochastic partial differential equation. Then, sufficient conditions are found on suboptimal controls in the form of a sot of non-linear partial differential equations. Finally, a central finite differences scheme is proposed to solve the set of equations, and the applicability of the scheme is demonstrated by examples.

Optimal control of a non-linear noisy sine wave oscillator

Abstract This paper deals with the optimal control of a random non-linear sine wave oscillator. It is assumed that the oscillator is subjected to two different kinds of perturbation. The first kind of perturbation is represented by a vector of independent standard Wiener processes and the second kind by a generalized type of a Poisson process. Sufficient conditions on the optimal controls are derived. These conditions are given by a set of two coupled non-linear partial integro-differential equations. A numerical procedure for the solution of these equations is suggested and its efficiency and applicability are demonstrated with examples.

On the Computation of Functionals Over Sample Paths of a Randomly Rotating Rigid Body

Abstract This paper deals with the computation of the values of two functionals which are defined over the sample paths of a randomly rotating rigid body. It is assumed that the body is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and the second kind by a homogeneous process with independent increments, finite second-order moments, mean zero and no continuous sample functions. In order to measure quantitatively the stochastic stability of the bodys motion, two functionals are defined over its sample paths. It is shown that each of these functionals is a solution to a corresponding partial integro-differential equation. A numerical procedure for the solution of these equations is suggested, and its efficiency and applicability are demonstrated with examples.