Necessary conditions for optimality for a distributed optimal control problem

A generalized reduced gradient method for the optimal control of multiscale dynamical systems

“…An indirect GRG solution method is presented in this section for computing the optimal macroscopic state and microscopic control trajectories for the DOC problem in (1)- (6). By this approach, a Lagrange multiplier, λ(x, t), is used to adjoin the dynamic and equality constraints, (5)- (8), (3), to the integral cost function (4), obtaining the augmented integral cost function,…”

“…Because analytical solutions to these PDEs are not available, this paper presents a GRG approach for reducing the computation required by the numerical solution of the DOC optimality conditions. The approach exploits the causality of the macroscopic dynamic equation (6) to representĴ solely as a function of u. Then an extremum of the DOC problem (1)-(6) can be found by determining the parameters of the control laws (10) that satisfy the optimality conditions.…”

“…On larger spatial and temporal scales, the agents' dynamics and interactions are assumed to give rise to macroscopic coherent behaviors, or coarse dynamics, described by partial differential equations (PDEs). This paper extends the capabilities of the DOC approach proposed in [6] for deterministic agent dynamics to agent dynamics that are governed by stochastic differential equations (SDEs). A new set of optimality conditions is derived, and a numerical indirect optimization approach is presented based on the GRG method [11].…”

“…Recently, the authors proposed a coarse-grained optimal control approach for large, multiscale dynamical systems, referred to as distributed optimal control (DOC), that enables the optimization of density functions, and/or their moments, subject to the agents' dynamic constraints [6]. The DOC approach in [6] is applicable to multiscale dynamical systems comprised of many agents or processes that, on small spatial and time scales, are each described by a small set of ordinary differential equations (ODEs), referred to as the microscopic or detailed equations.…”

“…The DOC approach in [6] is applicable to multiscale dynamical systems comprised of many agents or processes that, on small spatial and time scales, are each described by a small set of ordinary differential equations (ODEs), referred to as the microscopic or detailed equations. On larger spatial and temporal scales, the agents' dynamics and interactions are assumed to give rise to macroscopic coherent behaviors, or coarse dynamics, described by partial differential equations (PDEs).…”