On Averaging of a Class of “Non-Singularly Perturbed" Control Systems

In this paper, we consider the system

where $\epsilon>0$ is a parameter, the controls $u(t)$ are measurable functions taking values in ${\rm I\kern-1.99997ptR}^{k}$, and $A$ and $B$ are $m\times m$ and $m\times k$ matrices that satisfy the rank controllability condition:

The function $g:{\rm I\kern-1.99997ptR}^{k}\times{\rm I\kern-1.99997ptR}^{m}\times{\rm I\kern-1.99997ptR}^{n}\to{\rm I\kern-1.99997ptR}^{n}$ is assumed to be bounded

and satisfying the Lipschitz condition in $(y,z)$ uniformly with respect to $u\in{\rm I\kern-1.99997ptR}^{k}$.

If the parameter $\epsilon$ was small, the system (1)-(2) would belong to the class of the so-called singularly perturbed (SP) systems. SP systems are characterized by the decomposition of the state variables into groups that change their values with rates of different orders of magnitude (so that some of them can be considered as fast/slow with respect to others), which is due to the presence of the small singular perturbations parameter $\epsilon$. Such systems describe processes and interactions in disparate time scales, and they have been extensively studied in the literature (see, e.g., research monographs [5], [11], [12], [13] and surveys [6], [14], [15], [18]).

One of the approaches to SP control systems is the averaging method based on the analysis of asymptotic properties of the sets of time averages of the equation describing the dynamics of the slow state variables over the fast control-state trajectories considered on the intervals $[0,S]$ for large values of $S>0$. If the limit of these sets exists as $S$ tends to infinity, then (under certain additional conditions) it defines the right-hand-side of the differential inclusion, the solutions of which approximate the slow components of the solutions of the SP system when the singular perturbation parameter $\epsilon$ tends to zero (see [7], [8], [9], [10], [17] and also [1], [2], [3] for related developments).

In this paper, the parameter $\epsilon$ is not assumed to be small, and to distinguish it from the case of singular perturbations, we call system (1)-(2) non-singularly perturbed. We show that the approximation of the $z$-components of the solution of system (1)-(2) by the solution of a certain differential inclusion is still valid, but this result is no longer asymptotic and holds for any $\epsilon>0$. Such an approximation is possible due to the fact that the $y$-components of the state variables of the system (1)-(2) can change their values arbitrarily fast (see Lemma 1 in the next section) while the rates of change of the $z$-components are bounded by the constant $M_{g}$ (see (4)).

To state our main result, let us introduce the differential inclusion

where

with $\bar{co}$ standing for the closure of the convex hull of the corresponding set.

Let $T$ be an arbitrary positive number. Denote by $\mathcal{Z}_{T}(\epsilon)$ the set of the $z$-components of solutions of the system (1)-(2) considered on the interval $[0,T]$, that is,

By $\mathcal{Z}_{T}$ denote the set of solutions of the differential inclusion (5) considered on this interval. Note that, as can be readily understood,

The main result of the paper is the following theorem.

REMARK I. Theorem 1 resembles results establishing that the Hausdorff distance (induced by the uniform convergence metric) between the set of the slow components of solutions of a SP control system and the set of solutions of the differential inclusion, constructed by averaging of the slow subsystem over the controls and the corresponding solutions of the fast one, tends to zero when the singular perturbations parameter tends to zero (see [7], [8], [9], [10], [17]). As mentioned above, in contrast to these results, Theorem 1 is not of asymptotic nature. The equality (8) is valid for any $\epsilon>0$, including, e.g., $\epsilon=1$.

REMARK II. Note that the statement of Theorem 1 would look similar to that of Filippov-Wazewski theorem (see [4]) if $y(t)$ in (2) was another control (that is, an arbitrary measurable function taking values in ${\rm I\kern-1.99997ptR}^{m}$) instead of being the solution of (1).

Let us discuss an implication of Theorem 1 for optimal control. Consider the optimal control problem

where $G(\cdot)$ is a continuous function and $inf$ is taken over the controls $u(\cdot)$ and the corresponding solutions $(y_{\epsilon}(\cdot),z_{\epsilon}(\cdot))$ of system (1)-(2). Consider also the optimal control problem

where $inf$ is taken over measurable functions $(u(\cdot),y(\cdot))\in{\rm I\kern-1.99997ptR}^{k}\times{\rm I\kern-1.99997ptR}^{m}$ and the corresponding solutions $z(\cdot)$ of the system

(both $u(\cdot)$ and $y(\cdot)$ are playing the role of controls in this system).

Consider the system

Note that this system looks similar to the fast subsystem (1) but, in contrast to the latter, it evolves in the time scale $\tau=\frac{t}{\epsilon}$ ((13) will be referred to as the associated system). The following lemma is the key element of the subsequent analysis.

Let $y(\tau,u(\cdot),y)$ stand for the solution of the associated system (13) obtained with a control $u(\tau)$ and the initial condition $y(0)=y$. For a fixed $z$, denote by $V(S,y,z)$ the set of the time averages:

where the union is taken over all controls (that is, over all measurable functions $u(\cdot)$ taking values in ${\rm I\kern-1.99997ptR}^{k}$). The set of the time averages similar to (14) was introduced/used in [7], [8], [9], [10], [17], where it was shown that, as $S$ tends to infinity, it converges in the Hausdorff metrics to a convex and compact set (provided that the associated system satisfies certain controllability or stability conditions). It was also shown that it is this set that defines the right-hand-side of the differential inclusion, the solutions of which approximate the dynamics of the slow components of the SP control system.

The lemma below establishes that in the case under consideration the set ${\rm cl}(V(S,y_{0},z))$ (the closure of the set $V(S,y_{0},z)$) is equal to a convex and compact subset of ${\rm I\kern-1.99997ptR}^{m}$ for any $S>0$, and it also provides an explicit representation for this set.

Proof of Theorem 1. Due to (7), to prove the theorem it is sufficient to show that

This, in turn, will be shown, if we establish that, for any $\delta>0$, corresponding to any solution $z(t)$ of the differential inclusion (5), there exists a control $u_{\epsilon}(t)$, which, being used in the system (1)-(2), generates the solution $(y_{\epsilon}(t),z_{\epsilon}(t))$ that satisfies the inequality

To prove the latter statement, take an arbitrary solution $z(t)$ of (5), choose an arbitrary $\delta>0$ and construct the control $u_{\epsilon}(t)$ that insures the validity of (26).

Take $S\in(0,\frac{T}{\epsilon}]$ and partition the interval $\ [0,T]\$ by the points

where $\lfloor\cdot\rfloor$ is the floor function (that is, for any real $x$, $\lfloor x\rfloor$ is the maximal integer that is less or equal than $x$).

Let $v_{0}$ be the projection of the vector $\ \left(\epsilon S\right)^{-1}\int_{0}^{t_{1}}\frac{dz(t)}{dt}dt\$ onto the set $V(z_{0})$. That is,

By Lemma 2, ${\rm cl}\,V(S,y_{0},z_{0})=V(z_{0})$. Therefore, there exists a control $u_{0}(\tau)$ such that, being used in the associated system (13) on the interval $\ [0,\frac{t_{1}}{\epsilon}],\$ ensures that

where $y_{0}(\tau):=y(\tau,u_{0}(\cdot),y_{0})$.

Define the control $u_{\epsilon}(t)$ on the interval $[0,t_{1})$ by the equation

and show how this definition can be extended to the interval $[0,T]$.

Assume that the control $u_{\epsilon}(t)$ has been defined on the interval $[0,t_{l})$ and denote by $(y_{\epsilon}(t),z_{\epsilon}(t))$ the corresponding solution of the system (1)-(2) on this interval. Extend the definition of $u_{\epsilon}(t)$ to the interval $[0,t_{l+1})$ by following the steps:

(a) Define $v_{l}\$ as the projection of the vector $\ \left(\epsilon S\right)^{-1}\int_{t_{l}}^{t_{l+1}}\frac{dz(t)}{dt}dt\$ onto the set $V(z_{\epsilon}(t_{l}))$. That is,

(b) Define a control $u_{l}(\tau)$ such that, being used in the associated system (13) on the interval $\ [\frac{t_{l}}{\epsilon},\frac{t_{l+1}}{\epsilon}],\$ ensures that

where $y_{l}(\tau)$ is the solution of the associated system (13) obtained with the control $u_{l}(\tau)$ and with the initial condition $y_{l}(\frac{t_{l}}{\epsilon})=y_{\epsilon}(t_{l})$. The existence of such control follows the fact that, by Lemma 2, ${\rm cl}\,V(S,y_{\epsilon}(t_{l}),z_{\epsilon}(t_{l}))=V(z_{\epsilon}(t_{l}))$.

(c) Define the control $u_{\epsilon}(t)$ on the interval $[t_{l},t_{l+1})$ as equal to $u_{l}(\frac{t}{\epsilon})$. That is,

Proceeding in a similar way, one can extend the definition of the control $u_{\epsilon}(t)$ to the interval $[0,t_{N_{\epsilon S}})$. On the interval $[t_{N_{\epsilon S}},T]$, the control $u_{\epsilon}(t)$ can be taken to be equal to an arbitrary element $u$. Denote by $(y_{\epsilon}(t),z_{\epsilon}(t))$ the corresponding solution of the system (1)-(2) on the interval $[0,T]$.

Denote

Then

where $\ \tau_{l}=\frac{t_{l}}{\epsilon}=lS$ (see (27)), and

where $M_{g}$ is defined in (4). Also notice that

since

Let us verify that thus defined control ensures the validity of (26). Subtracting the equation

from the equation