Optimal Control Under Stochastic Uncertainty

Optimal control problems as arising in different technical (mechanical, electrical, thermodynamic, chemical, etc.) plants and economic systems are modeled mathematically by a system of first-order nonlinear differential equations for the plant state vector $$z=z(t)$$ z = z ( t ) involving, e.g., displacements, stresses, voltages, currents, pressures, concentration of chemicals, demands, etc. This system of differential equations depends on the vector u(t) of input or control variables and a vector $$a=a(\omega )$$ a = a ( ω ) of certain random model parameters. Moreover, also the vector $$z_0$$ z 0 of initial values of the plant state vector $$z=z(t)$$ z = z ( t ) at the initial time $$t=t_0$$ t = t 0 may be subject to random variations. While the actual realizations of the random parameters and initial values are not known at the planning stage, we may assume that the probability distribution or at least the occurring moments, such as expectations, variances, etc., are known. Moreover, we suppose that the costs along the trajectory and the terminal costs G are convex functions with respect to the pair (u, z) of control and state variables u, z, the final state $$z(t_f)$$ z ( t f ) , respectively. The problem is then to determine an open-loop, closed-loop, or an intermediate open-loop feedback control law minimizing the expected total costs consisting of the sum of the costs along the trajectory and the terminal costs. For the computation of stochastic optimal open-loop controls at each starting time point $$t_b$$ t b , the stochastic Hamilton function of the control problem is introduced first. Then, a H-minimal control can be determined by solving a finite-dimensional stochastic optimization problem for minimizing the conditional expectation of the stochastic Hamiltonian subject to the remaining deterministic control constraints at each time point t. Having a H-minimal control, the related Hamiltonian two-point boundary value problem with random parameters is formulated for the computation of the stochastic optimal state and adjoint state trajectory. In the case of a linear-quadratic control problem the state and adjoint state trajectory can be determined analytically to a large extent. Inserting then these trajectories into the H-minimal control, stochastic optimal open-loop controls are found. For approximate solutions of the stochastic two-point boundary problem, cf. [31].