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Consider a control system
![\begin{displaymath}
\dot{x}=f(x,u),
\end{displaymath}](img2.gif) |
(1.1) |
where controls are Lebesgue measurable functions
, with the control constraint set
being compact. The following hypotheses will remain in
effect throughout:
- (F1)
-
is continuous
and locally Lipschitz in the state variable
; that is, for each
bounded subset
of
, there exists
such that
whenever
and
are in
.
- (F2)
satisfies the linear growth condition
for some positive constants
and
.
Under these assumptions, for each control function
and
each initial phase
,
there exists a unique solution
on
satisfying
. Although the
preceding hypotheses on the dynamics can be relaxed in certain
results to follow, for ease of exposition we will retain the above
assumptions throughout the entire article.
A central issue in control theory is the construction of feedback control
laws which achieve a desired behavior of the control system (1.1) in a
manner which is both
- universal: the desired behavior achieved by the feedback holds
for all initial phases in a prescribed set
and
- robust: the feedback is effective in the presence of external
disturbance and state measurement error.
Two general types of feedback control problems will be considered in this
article; these can be described in very general terms as follows:
Some articles and general references which are relevant to the present work,
but which are not directly referenced below as are others, include [36], [48], [39], [40], [54], [2],
[3], [21], [25], [28], [29], [30], and [43] .
Our approach in this survey is to use the constructions employed
in solving the stabilizability problem, where an infinitesimal
decrease property of a nonsmooth control Lyapunov function is
utilized, in designing feedbacks in the optimal control problem;
there, the infinitesimal decrease of the value function is employed
in an analogous way.
Even in very simple cases, one cannot expect the existence of universal
feedback laws
which are continuous, this being the minimal condition for
the classical existence theory of ordinary differential equations to apply
to (1.2) or (1.3). This inadequacy of continuous feedback is
well illustrated via the following two examples.
Example 1.1 A famous result of Brockett [
7] asserts that when
![$f(\cdot ,\cdot )$](img34.gif)
is assumed to be
![$C^{1}$](img35.gif)
-smooth,
stabilizability by means of a continuous feedback law implies the
covering condition that the image of
![$f$](img14.gif)
contains an open
neighborhood of the origin; for an elegant degree-theory-based
proof of this fact, attributed to R. D. Nussbaum ca. 1982, see
Sontag [
50]. Brockett provided the following
3-dimensional example, known as the ``nonholonomic integrator,''
showing that the covering condition can fail for asymptotically
controllable bilinear systems:
The asymptotic controllability is verifiable by an ad hoc argument; the
failure of the covering condition is easy to check.
Remark 1.2 By a
Filippov solution (see [
24]) of
![% latex2html id marker 1312
$\dot{x}%
=f(x,k(x))=:g(x)$](img37.gif)
is meant an absolutely continuous function
![$x$](img6.gif)
solving the differential inclusion
where the second intersection is
taken over all subsets
![${\cal N}$](img39.gif)
of
![% latex2html id marker 1320
$\mathbb{R}^{n}$](img40.gif)
with Lebesgue
measure zero. The belief that the Filippov solution concept was
adequate for using discontinuous feedback laws to achieve
stabilizability was negated by a result of Ryan [
46]; see
also Coron and Rosier [
22]. On the other hand, the
problem of finding an adequate solution concept for discontinuous
feedback under which asymptotic controllability and feedback
stabilization are equivalent was first solved by Clarke, Ledyaev,
Sontag, and Subbotin [
13]. There,
as in the present exposition, a discretized solution concept was employed.
The main idea in the next example goes back to Barabanova and
Subbotin [4], [5]; see also Krasovskii
and Subbotin [35], [34] as well as Clarke,
Ledyaev, and Subbotin [16].
Example 1.3 Consider the optimal control problem with dynamics
where
![% latex2html id marker 1322
$x\in \mathbb{R}$](img42.gif)
,
![$u(\cdot )$](img18.gif)
is valued in
![$U=[-1,1]$](img43.gif)
, and
where it is desired to minimize the endpoint cost functional
![$\ell
(x(1))=-\vert x(1)\vert$](img44.gif)
. First consider any continuous feedback
![$k(t,x)$](img45.gif)
which is Lipschitz in
![$x$](img6.gif)
and such that solutions of (
1.3) are defined on
![$[0,1]$](img46.gif)
for every
![% latex2html id marker 1336
$%
\alpha \in \lbrack -1,1]$](img47.gif)
. Then there is a unique solution
![$x(t)=x(t;\alpha ) $](img48.gif)
of
![$\dot{x}=k(t,x)$](img49.gif)
on
![$[0,1]$](img46.gif)
for each such
![$\alpha $](img50.gif)
. Now note that the function
![$Q:[-1,1]\rightarrow \lbrack
-1,1]$](img51.gif)
given by
is continuous and, therefore, has a fixed point
![$\hat{\alpha}$](img53.gif)
by
virtue of Brouwer's fixed point theorem. Furthermore,
![$x(1;\hat{\alpha})=0$](img54.gif)
. Now consider the discontinuous feedback law
Upon substituting into the dynamics, we see that for
any
![$\alpha \in \lbrack -1,1]$](img56.gif)
, the resulting trajectory satisfies
![$-\Vert x(1)\Vert \leq -1$](img57.gif)
. This argument can be extended, by
approximation, to the case of merely continuous
![$k$](img33.gif)
. Therefore,
given any continuous feedback
![$k$](img33.gif)
, there is at least one initial
phase
![$(0,\hat{\alpha})$](img58.gif)
with
![$\hat{\alpha}\in \lbrack -1,1]$](img59.gif)
,
such that the above discontinuous feedback produces a better
outcome than the continuous one. Hence, there does not exist a
continuous feedback which is optimal universally for initial
phases in the set
![$\{0\}\times \lbrack -1,1]$](img60.gif)
. (Of course, it is
not always possible to obtain a classical solution to the
differential equation
when the feedback
![$k$](img33.gif)
is
discontinuous, since the existence theory of ordinary differential
equations can break down, but in the present example, this
difficulty does not occur.)
We now pause to fix our notation from nonsmooth analysis. Our main general
references for the basic results we will employ from this subject are
Clarke, Ledyaev, Stern, and Wolenski [20]; see also [14],
Clarke [9], [10], and Loewen [41].
The Euclidean norm is denoted
, and
is the usual inner product. The open unit ball in
is denoted
. For a set
, we denote by
,
,
,
and
the
complement, convex hull, closure, boundary, and interior of
, respectively.
Let
be a nonempty subset of
. The distance of a point
to
is given by
The metric projection of
on
is denoted
If
and
, then the vector
is called a
perpendicular to
at
. The cone consisting of all
nonnegative multiples of these perpendiculars is denoted
and is referred
to as the proximal normal cone (or P-normal cone) to
at
. If
or no perpendiculars to
exist at
, then we set
. Observe that the P-normal cone is a local
construction, since as can readily be shown,
Let
be an extended real
valued function which is lower semicontinuous; that is, for each
,
a property
equivalent to closedness of the epigraph of
,
We denote the effective domain of
by
A vector
is said to be a proximal subgradient (or P-subgradient)
of
at a point
provided that
The set of all such vectors is called the P-subdifferential of
at
, denoted
. One can show that
iff there exists
such that
for all
near
, and that
for a
dense subset of
(where we adopt the convention that
when
).
The limiting normal cone (or L-normal cone) to
at
is defined to be the set
The L-normal cone leads to a corresponding L-subdifferential set
for
:
the members of which are called L-subgradients.
The Clarke normal cone (or C-normal cone) to
at
is defined by
and the corresponding subdifferential for
at
is
The lower Dini derivate of
at
in the direction
is
![\begin{displaymath}
Dg(y;w) := \liminf_{\stackrel{\lambda\downarrow 0}{u\to w}}\frac{g(y+\lambda
u)-g(y)}{\lambda}.
\end{displaymath}](img110.gif) |
(1.4) |
In case
is locally Lipschitz, one has the simplification
![\begin{displaymath}
Dg(y;w) = \liminf_{\lambda\downarrow 0}\frac{g(y+\lambda w)-g(y)}{\lambda}.
\end{displaymath}](img111.gif) |
(1.5) |
The discretized solution concept we will work with includes a
state measurement error
and an external disturbance
in a
perturbed version of (1.1) under a (generally discontinuous)
feedback
, modeled by
Here
is a bounded measurable function,
but no measurability assumptions are made on
Let
be a partition of
, by which is meant a countable, strictly increasing
sequence
with
and
as
. Let an initial phase
be specified. The associated
-trajectory
is the curve
satisfying
and
![\begin{displaymath}
\dot{x}_{\pi }(t)=f(x_{\pi }(t),k(t_{i},x_{\pi }(t_{i})+p(t_{i})))+q(t)\quad
\forall \,t\in (t_{i},t_{i+1}),\quad i \geq 0.
\end{displaymath}](img128.gif) |
(1.6) |
Note that
is the unique solution on
of the
differential equation
satisfying
, with a piecewise constant control function
determined by the feedback
and the measurement error
. This
solution procedure, which involves discretizing the feedback
control law
via ``closed loop system sampling,'' is the same
as the ``step-by-step'' solution concept employed by Krasovskii
and Subbotin [34], [35] in differential game
theory.
Next: 2 Stabilizability
Up: clark
Previous: clark
2003-08-05