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Consider a control system

(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
is assumed to be
smooth,
stabilizability by means of a continuous feedback law implies the
covering condition that the image of
contains an open
neighborhood of the origin; for an elegant degreetheorybased
proof of this fact, attributed to R. D. Nussbaum ca. 1982, see
Sontag [
50]. Brockett provided the following
3dimensional 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
is meant an absolutely continuous function
solving the differential inclusion
where the second intersection is
taken over all subsets
of
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
,
is valued in
, and
where it is desired to minimize the endpoint cost functional
. First consider any continuous feedback
which is Lipschitz in
and such that solutions of (
1.3) are defined on
for every
. Then there is a unique solution
of
on
for each such
. Now note that the function
given by
is continuous and, therefore, has a fixed point
by
virtue of Brouwer's fixed point theorem. Furthermore,
. Now consider the discontinuous feedback law
Upon substituting into the dynamics, we see that for
any
, the resulting trajectory satisfies
. This argument can be extended, by
approximation, to the case of merely continuous
. Therefore,
given any continuous feedback
, there is at least one initial
phase
with
,
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
. (Of course, it is
not always possible to obtain a classical solution to the
differential equation
when the feedback
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 Pnormal cone) to at . If
or no perpendiculars to exist at , then we set
. Observe that the Pnormal 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 Psubgradient)
of at a point
provided that
The set of all such vectors is called the Psubdifferential 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 Lnormal cone) to at is defined to be the set
The Lnormal cone leads to a corresponding Lsubdifferential set
for :
the members of which are called Lsubgradients.
The Clarke normal cone (or Cnormal cone) to at
is defined by
and the corresponding subdifferential for at is
The lower Dini derivate of at in the direction is

(1.4) 
In case is locally Lipschitz, one has the simplification

(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

(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 ``stepbystep'' solution concept employed by Krasovskii
and Subbotin [34], [35] in differential game
theory.
Next: 2 Stabilizability
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20030805