Reachability Set

Abstract

For the linear control systems it is proved Cauchy formula, which represents the trajectory of the system with the help of the fundamental matrix. We list the properties of the fundamental matrix, introduce the notion of the reachability set of a linear system and establish its basic properties: the limitation, convex, closure, and continuity. It is showed the relation of a special family of extreme controls with the boundary of a reachability set.

Exercise Set

• 1. Let the range of control $U \subset R^{n}$ be symmetrical about a point $\bar{u}$, i.e., from the condition $\bar{u} + v \in U$, it follows $\bar{u} - v \in U$. Show that a reachability set $Q(t_{1} )$ of a linear system has the same symmetry property with respect to the point

  $$ \bar{x} = F(t_{1} ,t_{0} )x_{0} + \int\limits_{{t_{0} }}^{{t_{1} }} {F(t_{1} ,t)B(t)\bar{u}dt} . $$

• 2. Show that regardless of the convexity of the range of control $U \subset R^{n}$, a reachability set $Q(1)$ of the system

  $$ \dot{x} = b(u),\,\,x(0) = 0,\,\,u \in U $$

  is convex.

  Hint: if points $x^{1} ,x^{2} \in Q(1)$ correspond to controls $u^{1} (t),\,\,u^{2} (t)$ then for $0 < \lambda < 1$ a point $x = (1 - \lambda )x^{1} + \lambda x^{2}$ corresponds to control

  $$ u(t) = u^{1} \left( {\frac{t}{\lambda }} \right),\,\,t < \lambda ;\quad u(t) = u^{2} \left( {\frac{t - \lambda }{1 - \lambda }} \right),\,\,t \ge \lambda . $$

• 3. Check that from the convexity of a reachability set, the convexity of its closure follows.

• 4. Graph in the plane $x,\dot{x}$ a reachability set $Q(1)$ of a second-order linear differential equation

  $$ {\ddot x} + a_{1} \dot{x} + a_{2} x = u,\,\,x(0) = 0,\,\,\dot{x}(0) = 0,\,\,\left| u \right| \le 1 $$

  with constant coefficients $a_{1} ,a_{2}$ for different roots of an auxiliary equation.

  Hint: reduce the equation to its canonical form by choosing a suitable coordinate system, and then use the extreme principle. We can use Example 3.5 as a sample that corresponds to the case $a_{1} = a_{2} = 0$.

• 5. Is the following statement true or false? If we suppose the non-uniqueness of the maximum points of $u(t,c)$ in the regularity condition (3.20) on interval $T(c) \subset [t_{0} ,t_{1} ]$ for some directions c, then a reachability set $Q(t_{1} )$ will be closed in the class of the piecewise continuous controls.