Menahem Friedman

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.

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 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.

Computation of optimal controls for two classes of non-linear distributed parameter systems

Two sets of non-linear partial differential equations are dealt with. Each of these sets serves as sufficient conditions for optimum. A numerical procedure for solving these equations is suggested. The procedure is applied over a large class of problems and the results verify the applicability of the method.

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.

Suboptimal controls of a second-order system subjected to discontinuous random perturbations

This paper doala with tho suboptimal control of a stochastic second-order system. It is assumed that the system is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and tho second kind by a homogeneous process with independent increments, finite second-order moments, mean zero and no continuous sample paths. The stochastic system is approximated by a diffusion type system, and the optimal feedback control laws for the diffusion type system are chosen as the suboptimal controls for the given system. It is shown that in order to construct tho suboptimal controls, a non-linear partial differential equation and a linear partial integro-difforontial equation, have to be solved. A finite difference algorithm for the solution of these equations is proposed, and its efficiency and applicability are demonstrated with examples.

On the random rotational motion of a ship with a two position automatic pilot

In this paper a weak stability property of a random rotational motion of a ship with a two position automatic pilot, is dealt with. This is done by choosing two different sets of stochastic differential equations as mathematical models for the ships motion. For each model, a partial differential equation over a bounded domain in the state space is derived. The value of the solution to the PDE at a point P in the domain provides the conditional expectation of the first time that the systems trajectory in the state space hits the boundary given that P is the starting point of the trajectory. The partial differential equations derived for the two models are of two and three independent variables, respectively. Numerical procedures for solving these equations are suggested, and their efficiency and applicability are demonstrated with examples. For each model, a functional is defined on the PDE solutions. The values of the functionals serve as a measure of the systems stochastic stability.