Many analyses of controllers and associated proofs rely on existence and uniqueness of system trajectories in continuous time. A large class of control systems possess discontinuities in time which oftentimes complicates the analysis. Particular examples include step response, event-triggered systems and optimal control where time-discontinuous control laws are typical. For some of the recent applications, where the discussed phenomenon arises [1]-[3]. Consider, for instance, the problem of optimal control of a vibrating spring with an attached unit mass described by the differential equation
$ \ddot{x} + \dot{x} = u $ |
where
$ u^*(t) = {\rm sgn} ( \sin(t + \delta) ) $ |
for a certain parameter
Consider now the problem of maximizing the number of queens in an ant colony. The dynamics can be described as follows:
$ \begin{align*} \dot{w}&= a w u - b w \\ \dot{q}&= (1 - u)w \end{align*} $ |
where
$ u^*(t) = \begin{cases} 0, \quad t \le \frac{1}{b} \ln \left(\frac{a}{a-b}\right) \\ 1, \quad \text{otherwise.} \end{cases} $ |
In such examples, the right-hand side of the system dynamics becomes discontinuous in time and classical system trajectories do not exist (they may, however, exist in the so called extended sense, or Caratheodory sense, which will be discussed further). In practice, the optimal controller
The classical theorem on the existence of solutions in the extended sense is due to Caratheodory [5]-[7]. Recent studies in this field include topics such as well-posedness of Caratheodory solutions for bimodal piecewise affine systems [8], existence and uniqueness results for monotone non-increasing right-hand sides [9], and extended Caratheodory solutions for hybrid systems [10]. Further, when the right-hand side of the system dynamics is discontinuous in the state variable, the situation becomes more complicated, than only with time discontinuities, and such notions as Filippov solutions may come into place [11], [12]. The foundation of such solutions extensively utilizes the theory of differential inclusions which are also used in partial differential equations [13]. Particular applications range from discontinuous stabilization [14], sliding-mode control [15], [16] and optimal control [17]. An overview of Filippov and other generalized solutions may be found in [18]. The proofs of these classical results rely on compactness arguments, such as Arzela-Ascoli theorem [19], and certain fixed-point theorems, such as Schauder fixed-point theorem [20] and Kakutani fixed-point theorem [21], which are, unfortunately, not constructive and do not, generally, provide explicit computational procedures [22]. The current work suggests to address the existence and uniqueness of the trajectories of discontinuous systems in a constructive framework. Currently, only the case of time discontinuities is covered. However, the result can be extended onto the case of discontinuous feedback control provided that the controller implementation is considered in the sample-and-hold manner (see details in Section Ⅳ).
The main contribution of this work is thus a constructive theorem on existence and uniqueness of system trajectories in the extended sense. Schwichtenberg [23], and Ye [24] addressed the case of initial value problems where the right-hand side of the differential equation satisfies the Lipschitz condition (the corresponding classical theorem is originally due to Picard and Lindelöf, and the corresponding description may be found in [5]). The new theorem derived in this work is a constructive counterpart of the Caratheodory's existence and uniqueness theorem. Constructive analysis [25], which is done in intuitionistic logic [26], offers a suitable framework for the purposes of the current work. A related example application of constructive analysis to control theory was recently provided for the Lyapunov stability theory [27].
The remainder of the work is structured as follows: Section Ⅱ is concerned with the key aspects of constructive analysis needed for the constructive proof of the theorem which is given in Section Ⅲ.
Ⅱ. PRELIMINARIESIn this section, some key notions and theorems of constructive analysis, needed for the current work, are discussed. This section mostly follows [24] since it addresses differential equations and measure theory together which will be required in the next section. First, a function
$ \begin{align*} & \forall \varepsilon \in \mathbb{Q}_{>0}, c \in \mathbb{Q}, r \in \mathbb{Q}_{>0} , \forall x, y \in \{ z : |z - c| \le r \} \\ & | x - y | \le \omega(\varepsilon, c, r) \implies | f(x) - f(y) | \le \varepsilon. \end{align*} $ |
A modulus of continuity is an important certificate that every function in constructive analysis must have inside its definition. It allows computing bounds on the change of the argument that leads to a change of the function values within a prescribed bound. A function that possesses a modulus of continuity is clearly uniformly continuous. In case of
Theorem 1: Let
Here, absolutely continuous means that for any
Now, constructive integrable functions are introduced. An integrable function is a pair
$ \sum\limits_{n=0}^{\infty} \int \limits_{\mathbb{R}} |f_n(t)| dt < \infty $ |
and
$ f(t) = \sum\limits_{n=0}^{\infty} f_n(t) $ |
on the set
$ \text{dom}(f) = \left\{ t: \sum\limits_{n=0}^{\infty} |f_n(t)| < \infty \right\} $ |
called domain of
$ \int_{\mathbb{R}} f(t) dt = \sum\limits_{n=0}^{\infty} \int_{\mathbb{R}} f_n(t) dt. $ |
Basic properties of the Lebesgue integral can be proven constructively. Integrable functions allow introducing characteristic functions of finite intervals. For example, the characteristic function
$ g_n(t):=\begin{cases} 0, &t \in (-\infty, 0] \cup [1, \infty) \\ nt, &t \in \left[0, \frac{1}{n} \right] \\ 1, &t \in \left[ \frac{1}{n}, 1 - \frac{1}{n} \right] \\ n(1-t), &t \in \left[ 1 - \frac{1}{n}, 1 \right] \end{cases} $ |
by the representation
Further, measurable functions are introduced (for details, please refer to [28, Chapter. 6] or [24, Chapter. 6]). First, a measure of a finite interval
$ \begin{equation} \sum\limits_{j=1}^K \chi_{[t_j, \tau_j]}(t) f_j(t) \label{eqn:simple-fncs} \end{equation} $ | (1) |
where each
Now, consider convergence of measurable functions. The key type of convergence used in the next section is convergence almost uniformly: a sequence
Lemma 1: [24, pp. 165] For any almost uniformly Cauchy sequence of measurable functions
In the next section, a constructive variant of Theorem 1 is addressed.
Ⅲ. RESULTS AND DISCUSSIONThe goal of this section is to obtain a constructive counterpart of Theorem 1 which addresses numerical uncertainty. The following theorem is suggested as a particular substitute and uses assumptions which can be justified from the practical standpoint. That is, the theorem requires that the right-hand side of the differential equation have a finite number of separable discontinuities. It is easy to show that in case of indistinguishable discontinuities, the proof would imply decidability of equality over real numbers which is not true constructively. The details are given below.
Theorem 2: Consider the initial value problem
$ \begin{equation} \dot{x} = f(x, t), \quad x(0)=x_0 \label{eqn:general-sys} \end{equation} $ | (2) |
on the rectangle
$ \begin{equation} f(x, t) = \sum\limits_{j=1}^{K} \chi_{[\tau_j, \tau_{j+1}]}(t) f_{j}(x, t) \label{eqn:rhs} \end{equation} $ | (3) |
such that
Proof: Denote dom
1)
2)
3) each subinterval may contain at most one
Condition 3 is not inevitable; it is only used to simplify the proof as will be shown below. Fix some arbitrary
$ \begin{equation} \cfrac{1}{N} \le \cfrac{\min \left\{ \frac{1}{\alpha} \omega_f \left( \frac{1}{n} \right), \frac{1}{F \alpha} \omega_f \left( \frac{1}{n} \right), \frac{2}{n} \right\}}{2}. \label{eqn:N-ineq} \end{equation} $ | (4) |
Inequality (4) is used in (5) to guarantee that (6), characterizing the precision of the solution, holds. That is, the constructed solution will satisfy the differential equation up to the precision of
$ \begin{align*} & \varphi_{n}(0) := x_0 \\ & \varphi_n(t) := \varphi_n(t_i) + f(\varphi_n(t_i), t_i)(t - t_i) \\&\forall t \in [t_i, t_{i+1}] \cap {\rm dom}_t(f). \end{align*} $ |
It follows that
$ \begin{equation} |\varphi_n(t) - \varphi_n(t_i)| \le \frac{F \alpha}{N} \le \cfrac{\omega_f \left( \frac{1}{n} \right) }{2} \end{equation} $ | (5) |
whence
$ \dot{\varphi}(t) = f(\varphi_n(t_i), t_i), \quad t \in I_N. $ |
Notice that
$ \mu (J_N) = \sum\limits_{\sigma(i)} |t_{\sigma(i)+1}-t_{\sigma(i)}| \le K \frac{\alpha}{N}. $ |
Further, for each
$ \begin{equation} |\dot{\varphi}_n(t) - f(\varphi_n(t), t)| = |f(\varphi_n(t_i), t_i) - f(\varphi_n(t), t)| \le \frac{1}{n}. \label{eqn:bound} \end{equation} $ | (6) |
from which it follows that
$ \int _{t_1}^{t} |\dot{\varphi}_n(\tau) - f(\varphi_n(\tau), \tau)| d \tau \le \int _{t_1}^{t} \frac{1}{n} d \tau = \frac{t - t_i}{n}. $ |
But
$ \begin{align*} & \left|\int _{t_1}^{t} \dot{\varphi}_n(\tau)-f(\varphi_n(\tau), \tau)d\tau \right| \\ &~~~~\le \int _{t_1}^{t} |\dot{\varphi}_n(\tau)-f(\varphi_n(\tau), \tau)| d\tau \le \frac{t-t_i}{n}. \end{align*} $ |
Furthermore, for each
$ \begin{align*} & \left|\int _{t_1}^{t} \dot{\varphi}_n(\tau)-f(\varphi_n(\tau), \tau) d\tau \right| \\ &~~~~\le \int _{t_1}^{t} |\dot{\varphi}_n(\tau)-f(\varphi_n(\tau), \tau)| d\tau \le 2 F \frac{\alpha}{N} \end{align*} $ |
holds since
$ \begin{align*} &\left| \varphi_n(t) - x_0 - \int _{0}^{t} f( \varphi_n (\tau) , \tau ) d \tau \right| \le \frac{\alpha}{n} + 2KF \frac{\alpha}{n} \\ &~~~~~~~~~~=\alpha ( 1 + 2KF ) \frac{1}{n}. \end{align*} $ |
Let
$ \begin{align*} &|\varphi_m(t) - \varphi_n(t)| \le \int _{0}^{t} | f(\varphi_m(\tau), \tau) - f(\varphi_n(\tau), \tau) | d \tau \\ &~~~~~~~~~~~+\alpha ( 1 + 2KF )\left(\frac{1}{m} + \frac{1}{n}\right) \\ &~~~~~~~~~\le\! L\! \int _0^{t} | \varphi_m(\tau)\!\! -\!\! \varphi_n(\tau) | d \tau\!\! +\!\! \alpha ( 1\!\! +\!\! 2KF )\left(\frac{1}{m}\!\! +\!\! \frac{1}{n}\right). \end{align*} $ |
Let
$ g(t) := \int _{0}^{t} \left| \varphi_m(\tau) - \varphi_n(\tau) \right|d \tau $ |
and
$ \varepsilon := ( 1 + 2KF )\left(\frac{1}{m} + \frac{1}{n}\right). $ |
It follows that:
$ \dot{g}(t) \le L g(t) + \varepsilon\alpha. $ |
Therefore,
$ \int_{0}^{t} \left( \dot{g}(\tau)e^{-L\tau} - Lg(\tau)e^{-L\tau} \right) d\tau \le \int_{0}^{t} \varepsilon \alpha e^{-L \tau} d \tau. $ |
Further,
$ |\varphi_m(t)-\varphi_n(t)| \le Lg(t) + \varepsilon \alpha \le \varepsilon\alpha e^{Lt} \le \varepsilon \alpha e^{L \alpha} $ |
since, as from the integral above
$ g(t) \le \frac{\varepsilon\alpha}{L} \left( e^{Lt}-1\right). $ |
At this point, the condition holds that for any
$ \lim \limits_{n \rightarrow \infty} (\varphi_n(t_i)+f(\varphi_n(t_i), t_i)(\tau_j-t_i)) = \lim\limits_{n \rightarrow \infty} \varphi_n(t_{i+1}). $ |
Since
$ |\psi(t) - \varphi_n(t)| \le L \int _{0}^{t} |\psi(t) - \varphi_n(t)| d \tau + \frac{\alpha}{n}( 1 + 2KF ) $ |
where
$ \psi(t) = x_0 + \int_{0}^{t} f(\psi(\tau), \tau)d\tau. $ |
Therefore
$ |\psi(t) - \varphi_n(t)| \le \frac{1}{n} \alpha ( 1 + 2KF ) e^{L \alpha}. $ |
That means that
Remark 1: It follows that the Picard-iteration
$ \begin{align*} & \varphi_0(t) := x_0\\ & \varphi_{n+1}(t) := x_0 + \int_{0}^{t} f(\varphi_n(\tau), \tau) d \tau \end{align*} $ |
is constructively well-behaved for the presented special case of
Remark 2: Uniform continuity of the solutions depending on the initial condition can be shown as in the standard case [24, pp. 106].
Remark 3: The assumption that the time discontinuities be rational numbers can be relaxed if there is a minimal gap between them which is known beforehand.
Remark 4: Theorem 2, unlike Theorem 1, requires the time discontinuities
One may see how a failure to meet the assumption in the last remark leads to malfunctioning of the algorithm in Theorem 2. Consider, for example, sliding-mode control where an ideal trajectory is supposed to come onto a sliding surface, but, in implementation, a phenomenon called "chattering" occurs when the numerically computed trajectory jumps back and forth around the sliding surface. This chattering may well depend on the chosen sample time. It can be seen how such a case would become problematic in the method of proof in Theorem 2. Classically, one considers other types of solutions such as, for example, in the sense of Filippov [29]. Constructive treatment of such cases might be a challenging task. However, there exist notions of trajectories of differential equations with discontinuous right-hand side which are based on the Caratheodory's solutions, such as sample-and-hold [18]. Theorem 2 is considered as a particular constructive counterpart of Caratheodory's Theorem 1. It covers a large class of functions in the right-hand side of initial value problems which have a finite number of separable discontinuities. It can be seen that the method used in the proof is not constructively applicable to the case if discontinuities are arbitrary real numbers since it would lead to decidability of equality over reals. However, it is suggested that all practical problems satisfy the conditions stated. The result can also be generalized to the case where
In this section, some types of dynamical systems for which Theorem 2 applies are discussed. First, the simplest case is the one in which the right-hand side of (2) possesses only one discontinuity in time. For example, consider the problem of optimal consumption in simple economy. The economy can be given by the following dynamics:
$ \begin{align*} &\dot{x} = u(t)x(t), \quad t \in [0, T], \quad T > 0 \\ &x(0) = x_0 \end{align*} $ |
where
$ P(u(\cdot)):= \int _{0}^{T} (1-u(t))x(t) dt. $ |
According to the Pontryagin's Maximum Principle, the optimal control equals
$ u^*(t)=\begin{cases} 1, \quad \text{ if } 0\le t \le t^* \\ 0, \quad \text{ if } t^* \le t \le T \end{cases} $ |
where
$ \begin{align*} &\min\limits_{\tau_{ijk, l_{ijk}}, u_{ijk}} \int_{0}^{T} \sum\limits_{i, j, k} u_{ijk}(t) P(s_k) dt \\ &{\text s.t.} \dot{x}_i = - \sum\limits_{j, k} s_k u_{ijk}, \forall i \\ &u_{ijk}(t) \in \{0, 1\}, \forall t \in [\tau_{ijk, l_{ijk}}, \tau_{ijk, l_{ijk}+1}], \forall i, j, k, l_{ijk} \\ &\tau_{ijk, l_{ijk}+1} - \tau_{ijk, l_{ijk}} \ge d, \forall i, j, k, l_{ijk} \\ &\tau_{ijk, l_{ijk}} - \tau_{i'j'k', l_{i'j'k'}} \ge d, \forall i, i', j, j', k, k', l_{ijk}, l_{i'j'k'} \end{align*} $ |
where
$ \begin{align*} &\dot x = f(x(t), \kappa(x(\tau_i), \tau_i), t \in [\tau_i, \tau_{i+1}] \\ &x(0) = x_0. \end{align*} $ |
In this case, the right-hand side takes the form:
$ \sum\limits_{i=1}^{N-1} \chi_{[\tau_i, \tau_{i+1}]}(t) f_i(x, t) $ |
where
$ \dot x = A_{\sigma(t)} x(t). $ |
Here, a piecewise constant function
This works was concerned with analysis of the Caratheodory's theorem on existence and uniqueness of solutions to discontinuous initial value problems within constructive mathematics. A particular variant of the theorem was formulated and proven that covers a large class of practical problems. One of the future important topics is a constructive framework for other generalized solutions, such as in the Filippov sense. However, even though the case of systems discontinuous in the state variable is not addressed by the Caratheodory's theorem, if the trajectories are to be considered in the sample-and-hold framework, the new result may be applied. In this regard, it is worthwhile to investigate constructive content of system stability under sample-and-hold control.
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