This work focuses on optimal quadratic control of a class of hybrid and switching systems. In the first part of this dissertation, we explore the effect of stochastically varying environments on the gene regulation problem. We use a mathematical model that combines stochastic changes in the environments with linear ordinary differential equations describing the concentration of gene products. Motivated by this problem, we study the quadratic control of a class of stochastic hybrid systems for which the lengths of time that the system stays in each mode are independent random variables with given probability distribution functions. We derive a sufficient condition for finding the optimal feedback policy that minimizes a discounted infinite horizon cost. We show that the optimal cost is the solution to a set of differential equations with unknown boundary conditions. Furthermore, we provide a recursive algorithm for computing the optimal cost and the optimal feedback policy.

When the time intervals between jumps are exponential random variables, we derive a necessary and sufficient condition for the existence of the optimal controller in terms of a system of linear matrix inequalities. In the second part of this monograph, we present the problem of optimal controller initialization in multivariable switching systems. We show that by finding optimal values for the initial controller state, one can achieve significantly better transient performance when switching between linear controllers for a not necessarily asymptotically stable MIMO linear process. The initialization is obtained by performing the minimization of a quadratic cost function. By suitable choice of realizations for the controllers, we guarantee input-to-state stability of the closed-loop system when the average number of switches per unit of time is smaller than a specific value.

If this is not the case, we show that input-to-state stability can be achieved under a mild constraint in the optimization.