# Sanjay Lall

- Professor of Electrical Engineering
- Professor of Aeronautics and Astronautics
- Information Systems Laboratory
- Stanford University

### Explicit Solutions to Separable Problems in Optimal Cooperative Control

This paper presents explicit state-space solutions for a class of networked cooperative control problems. We consider the optimal control problem for a class of systems with dynamics defined at the vertices of a directed acyclic graph, and dynamic propagation along the edges of the graph. The controller is also constrained to use information flowing along the edges of the graph. With the additional assumption of transitive closure, such problems have linear optimal controllers. We show that, under the factorization condition that every disturbance non-sink takes state feedback, these problems separate into several independent subproblems and hence they can be explicitly solved. This technique generalizes and unifies several existing approaches for optimal decentralized control, and it further provides the optimal controller for many cases in which the optimal control was previously unknown. The optimal solutions are expressed in terms of a series of Riccati equations whose sizes are bounded by the system order. The separated problems highlight the structure of the optimal controller. We demonstrate our approach on a salvo guidance problem for a team of networked missiles.The code for the numerical examples.