Semidecidable Controller Synthesis for Classes of Linear Hybrid Systems

A problem of great interest in the control of hybrid systems is the design of least restrictive controllers for reachability specifications. Controller design typically uses game theoretic methods to compute the region of the state space for which there exists a control such that for all disturbances, an unsafe set is not reached. In general, the computation of the controllers requires the steady state solution of a Hamilton-Jacobi partial differential equation which is very difficult to compute, if it exists. In this paper, we show that for special classes of hybrid systems where the continuous vector fields are linear, the controller synthesis problem is semi-decidable: There exists a computational algorithm which, if it terminates in a finite number of steps, will exactly compute the least restrictive controller. This result is achieved by a very interesting interaction of results from mathematical logic and optimal control.