Abstract
For a system of two nonlinear differential equations $$\frac{dx}{dt}=f(x)+\nu u$$, the question of the reachability of the origin is considered. Here $$x=(\xi,\eta),\quad\frac{dx}{dt}=\biggl(\frac{d\xi}{dt},\frac{d\eta}{dt}\biggr),\quad f(x)=(f_1(\xi,\eta),f_2(\xi,\eta))$$ are variable vectors of the phase plane $R^2$, $u=(u_1,u_2)$ is a constant vector of the plane $R^2$, and $\nu(t)$ is a piecewise continuous scalar function called an admissible control and satisfying the condition $|\nu(t)|\le1$. The paper provides definitions of reachability and non-reachability of the origin in the small for the system under consideration and proves three theorems formulating sufficient conditions for the existence of a reachable set. In conclusion, three examples are presented. The first example shows that the origin is unreachable in the small, although the system is asymptotically stable in the large; in the second example, the origin is unreachable in the small, but a reachable set exists; the third example is of interest because for nonlinear systems, the reachable set may be non-convex. 2 illustrations. 9 bibliography entries.
Full Text
Introduction
In 1967, I. P. Korolev investigated the dynamics of a system described by the differential equation:
$$\begin{aligned} \frac{dx}{dt} = f(x) + vu \end{aligned}$$
where $x \in D \subset R^2$, $u$ is a control parameter, and $|v| \le 1$. This work builds upon the foundational theories established in \cite{1, 2, 3, 4, 5} and further developed in \cite{6, 7}. The analysis focuses on the behavior of the system trajectories near the origin $x = 0$ and the properties of the switching surface $S(0)$.
The stability and topological structure of the phase space are determined by the function $f(x)$ and the control $v$. Specifically, we consider the case where $f(x)$ is a smooth function belonging to the class $C^k$. The equilibrium point at the origin is analyzed by examining the neighborhood $V(0)$ and the corresponding stability region $S(0)$. For a given initial point $x_0 \in V(0)$, the trajectory $x(x_0, t)$ is governed by the control law $v(t)$, which ensures that the system remains within the boundaries of $S(0)$.
Analysis of the Switching Surface
To characterize the behavior of the system, we introduce the transformation to coordinates $(\xi, \eta)$. The system (1) can be rewritten in the following form:
$$\begin{aligned} \frac{d\eta}{d\xi} = f_a(\xi, \eta) + u \end{aligned}$$
where $f_a$ represents the transformed vector field. Following the methodology in \cite{8}, we define a sequence of coefficients $d_s$ to analyze the local geometry of the trajectories:
$$\begin{aligned} d_2 &= D_1^{1/2}(1, k) - k f_1(1, k) \ d_3 &= D_2^{1/2}(1, k) - k D_2 f_1(1, k) \end{aligned}$$
If $d_2 = 0$, then $d_3$ determines the curvature of the switching line. In general, for $s = 2, 3, \dots$, we define:
$$\begin{aligned} d_s = D_{s-1}^{1/2}(1, k) - k D_{s-1} \end{aligned}$$
If $d_1 = d_2 = \dots = d_{2s-1} = 0$ and $d_{2s} \neq 0$, the system exhibits specific bifurcations near the switching surface $S(0)$ when $v = \pm v_0$.
Geometric Properties of Trajectories
As shown in [FIGURE:1], the switching surface $S(0)$ divides the phase space into regions $S^+$ and $S^-$. For $t > 0$, the trajectories $x(x_0, t)$ originating in $S^+$ move towards the boundary $f_+$, while those in $S^-$ move towards $f_-$. According to the criteria established in \cite{9}, when $v = 1$, the trajectories enter the region $S^-$, and when $v = -1$, they transition into $S^+$.
The intersection of these trajectories forms a closed loop, denoted as $AMLBNPA$ in the neighborhood $V(0)$. For any $x_0 \in V(0)$, the control law $v(t)$ ensures that the trajectory remains within the region $S(0)$. This behavior is illustrated in [FIGURE:2], which depicts the phase portrait for the case where $f(x)$ is a cubic polynomial and $v$ is not constant.
Numerical Examples and Results
Consider the system where $f(x)$ is defined such that:
$$\begin{aligned} \frac{d\xi}{dt} = \xi + v, \quad \frac{d\eta}{dt} = 3\xi + 4\eta + 4\xi^3 - v \end{aligned}$$
In this case, the equilibrium points are located at $(-1, 2)$ and $(-v, v + v^3)$. For $v = 1$ and $v = -1$, the trajectories exhibit distinct behaviors. Specifically, for $v = -1$, the switching line is defined by the relation $\eta = \xi^2$ for $\xi < 0$. As $t$ increases, the trajectories converge toward the stable manifold, confirming the theoretical predictions regarding the structure of $S(0)$.
These results extend the classical findings of L. I. Rozonoer and others \cite{1, 2, 3} regarding the optimal control of second-order systems. The proposed method for analyzing the coefficients $d_s$ provides a robust framework for determining the stability of nonlinear control systems with switching boundaries.
References
- E. F. Mishchenko, Prikladnaya Matematika i Mekhanika (PMM), Vol. 23, No. 2, pp. 209–229, 1959.
- V. V. Korolev, Izvestiya Vuzov, Matematika, No. 6, pp. 81–87, 1959.
- N. N. Krasovskii, Matematicheskiy Sbornik, Vol. 20, No. 3, pp. 153–174, 1965.
- V. G. Boltyanskii, Mathematical Methods of Optimal Control, Nauka, 1966.
- R. Bellman, Dynamic Programming, Princeton University Press, 1957.
- V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Gostekhizdat, 1949.
- I. P. Korolev, Uchenye Zapiski MGU, No. 12, pp. 86–117, 1967.