Differential inclusions and quasi-Lyapunov functions

TL;DR

The paper proves a sufficient condition for the existence of solutions to differential inclusions $\dot{x} \in F(x)$ with uniformly bounded, nonempty closed (potentially nonconvex) values by developing invariant partial $\varepsilon$-approximations and a piecewise smooth quasi-Lyapunov function that drives trajectories away from 'bad points'. The main result shows that $\varepsilon$-solutions exist from boundary points of the 'bad set' when a quasi-Lyapunov function exists, and, under successive refinements and $\varepsilon \to 0$, a true solution emerges, yielding an Olech-type corollary. The method extends patchy vector-field ideas to nonconvex right-hand sides by using weak forward invariance of partition elements and a Lyapunov-like mechanism to control the approximation process. An illustrating Fuller problem example demonstrates the applicability, constructing a semi-algebraic quasi-Lyapunov function and a time-augmented invariant approximation to guarantee existence of a solution starting at the origin, highlighting the practical relevance for controllable systems with chattering.

Abstract

A sufficient condition for existence of a solution of a differential inclusion with a uniformly bounded right-hand side that has nonempty closed (possibly nonconvex) values is obtained. An Olech-type result is obtained as a corollary. An example, which originates from the Fuller problem from optimal control theory, is given to demonstrate the applicability of the main result.

Figures (2)

• Figure 1: This figure is an attempt to give some intuition on the construction of the elements of the relatively open partition $\mathcal{V}^s$ of $\widetilde{D}$ which are contained in the first strip. Let us assume that the first strip is contained in the union of $\tilde{z}_i + K_{\delta_i}$, $i=1,2,3$. The "ice-cream cones" $\widetilde{z_1}+K_{\delta_1}$ and $\widetilde{z_2}+K_{\delta_2}$ are entirely contained in $\widetilde{D_1}\setminus\widetilde{C_1}$, so we can set $Z_1 = \widetilde{z_1}+K_{\delta_1}$ and $Z_2 = \widetilde{z_2}+K_{\delta_2}$. The "ice-cream cone" $\widetilde{z_3}+K_{\delta_3}$ intersects $\widetilde{C_2}$, so we split it into $\left(\widetilde{z_3}+K_{\delta_3}\right) \cap \left(\widetilde{D_1} \setminus \widetilde{C_1}\right)$ and $\left(\widetilde{z_3}+K_{\delta_3}\right) \cap \left(\widetilde{D_2} \setminus \widetilde{C_2}\right)$. Since a trajectory of $\widetilde{F}$ can pass from $\left(\widetilde{z_3}+K_{\delta_3}\right) \cap \left(\widetilde{D_2} \setminus \widetilde{C_2}\right)$ to $\left(\widetilde{z_3}+K_{\delta_3}\right) \cap \left(\widetilde{D_1} \setminus \widetilde{C_1}\right)$ through $\widetilde{C_2}$ within this strip, we set $Z_3 = \left(\widetilde{z_3}+K_{\delta_3}\right) \cap \left(\widetilde{D_2} \setminus \widetilde{C_2}\right)$ and $Z_4 = \left(\widetilde{z_3}+K_{\delta_3}\right) \cap \left(\widetilde{D_1} \setminus \widetilde{C_1}\right)$. In this way, we obtain all the elements $V_j^s, j \in \{1,\dots,5\}$ of $\mathcal{V}^s$ which are contained in the first strip. A trajectory of $\widetilde{F}$ can leave each of these sets by moving to a set with a greater number or by moving to the next strip. Moreover, a trajectory of $\widetilde{F}$ can leave the set $V_5^s$ only by moving to the next strip because of the choice of the cones' heights.
• Figure 2: The construction of the parabola $\pi$ (the dotted curve). The black curves are trajectories of the inclusion.

Theorems & Definitions (17)

• Definition 2.1: KR
• Definition 2.2: CLSW
• Definition 2.3: AC, CLSW
• Theorem 2.4: AC
• Definition 2.5: NR
• Definition 2.6: KR
• Definition 2.7: KR
• Definition 2.8: KR
• Remark 2.9: KR
• Proposition 2.10