7.4.1 Hybrid optimal control problem

We begin by defining the class of hybrid control systems and associated optimal control problems that we want to study. The first ingredient of our hybrid control system is a collection of (time-invariant) control systems

$$\dot x=f_q(x,u),\qquad q\in Q$$ (7.32)

where $Q$ is a finite index set; for simplicity, we assume that all the systems in the collection (7.32) share the same state space $\mathbb{R}^n$ and control set $U\subset \mathbb{R}^m$ . The second ingredient is a collection of switching surfaces (or guards) $S_{q,q'}\subset \mathbb{R}^{2n}$ , one for each pair $(q,q')\in Q\times Q$ . A function $x:[t_0,t_f]\to\mathbb{R}^n$ is an admissible trajectory of our hybrid system corresponding to a control $u:[t_0,t_f]\to U$ if there exist time instants

$$t_0<t_1<\dots<t_k<t_{k+1}:=t_f$$

and indices $q_0,q_1,\dots,q_k\in Q$ such that $x(\cdot)$ satisfies

$$\dot x(t)=f_{q_i}(x(t),u(t))\qquad \forall\,t\in(t_i,t_{i+1}), \ i=0,1,\dots,k$$ (7.33)

and

$$\begin{pmatrix}x(t_i^-)\\ x(t_i^+)\end{pmatrix}\in S_{q_{i-1},q_i},\qquad i=1,\dots,k.$$

Here $x(t_i^-)$ and $x(t_i^+)$ are the values of $x$ right before and right after $t_i$ , respectively, and the value $x(t_i)$ is taken to be equal to one of these one-sided limits, depending on the desired convention. At each possible discontinuity $t_i$ , a discrete transition (or switching event) is said to occur. The function $q:[t_0,t_f]\to Q$ defined by $q(t):=q_i$ for $t\in[t_i,t_{i+1})$ describes the evolution of $q$ along the trajectory; $q$ is often called the discrete state of the hybrid system. We can use it to rewrite (7.33) more concisely as $\dot x(t)=f_{q(t)}(x(t),u(t))$ for $t\ne t_i$ , $i=1,\dots,k$ .

We consider cost functionals of the form

$$J(u,\{t_i\},\{q_i\}):=\sum_{i=0}^k \int_{t_i}^{t_{i+1}}L_{q_i}(x(t),u(t))dt +\sum_{i=1}^k\Phi_{q_{i-1},q_i}(x(t_i^-),x(t_i^+))$$ (7.34)

where $L_q:\mathbb{R}^n\times U\to\mathbb{R}$ is the usual running cost and $\Phi_{q,q'}:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ is the switching cost, for $q,q'\in Q$ . For simplicity, we do not include a terminal cost (this is no loss of generality, as terminal cost can be easily incorporated into the above running-plus-switching cost along the lines of Section 3.3.2). We also introduce an endpoint constraint, characterized by a set $E_{q,q'}\subset \mathbb{R}^{2n}$ for each pair $(q,q')\in Q\times Q$ , according to which the trajectory $x(\cdot)$ must satisfy

$$\begin{pmatrix}x(t_0)\\ x(t_f)\end{pmatrix}\in E_{q_0,q_k}.$$ (7.35)

(Here $E_{q_0,q_k}$ plays the same role as $S_2$ at the end of Section 4.3.1.) Then, the hybrid optimal control problem consists in finding a control that minimizes the cost (7.34) subject to the endpoint constraint (7.35). We emphasize that the choice of a control $u$ is accompanied by the choice of two finite sequences $\{t_i\}$ and $\{q_i\}$ , to which we henceforth refer as the time sequence and switching sequence, respectively.