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Stability of scheduled sub-optimal feedback control

We present a general approach to proving the stability of gain-scheduled control laws when the gains are obtained as the solution of a soft-constrained time-optimal control problem. Usually gains can be computed off-line and stored but the optimum value of the scheduling variable, here the time to go, must be computed online, which is generally problematic. The stability proof allows to have stable scheduled gains without the need to compute the optimum time to go. The scheduling can be very simple and/or quasi-optimal. Remarkably, if the system matrix is nonsingular and the optimal gains are used, then the origin of the closed-loop system remains asymptotically stable even when the scheduling is sloppy.