Optimal Control Under Stochastic Uncertainty

Optimal control and regulator problems that arise in many concrete applications (mechanical, electrical, thermodynamical, chemical, etc.) are modeled by dynamical control systems obtained from physical measurements and/or known physical (a priori) laws. The basic control system (input–output system) is mathematically represented by a system of first order differential equations with random parameters: ż ( t ) = g ( t , ω , z ( t ) , u ( t ) ) , t 0 ≤ t ≤ t f , ω ∈ Ω z ( t 0 ) = z 0 ( ω ) . $$\displaystyle \begin {array}{rcl} \dot z (t) & = & g \Big ( t, \omega , z (t) , u (t) \Big ), t_0 \leq t \leq t_f , ~ \omega \in \Omega \\ z (t_0) & = & z_0 ( \omega ). \end {array} $$ Here, ω is the basic random element taking values in a probability space ( Ω , A , P ) $$(\Omega , \mathcal {A}, P)$$ , Probability space and describing the random variations of model parameters or the influence of noise terms. The probability space ( Ω , A , P ) $$(\Omega , \mathcal {A}, P)$$ consists of the sample space or set of elementary events Ω, the σ-algebra A $$\mathcal {A}$$ of events and the probability measure P. The plant state vector z = z(t, ω) is an m-vector involving direct or indirect measurable/observable quantities like displacements, stresses, voltage, current, pressure, concentrations, etc., and their time derivatives (velocities), z 0(ω) is the random initial state. The plant control or control input u(t) is a deterministic or stochastic n-vector denoting system inputs like external forces or moments, voltages, field current, thrust program, fuel consumption, production rate, etc. Furthermore, ż $$\dot z$$ denotes the derivative with respect to the time t.