Download Nonconvex Optimal Control Problems with Nonsmooth Mixed State

Nonconvex Optimal Control Problems
with Nonsmooth Mixed State Constraints
Maria do Rosário de Pinho,
ISR and DEEC, Faculdade de Engenharia da Universidade do Porto,
Rua Dr. Roberto Frias , 4200-465 Porto,Portugal
Email: [email protected]
Geraldo Nunes Silva
Depto de Ciências de Computação e Estatı́stica, IBILCE, UNESP,
15054-000, São José do Rio Preto, SP
E-mail: [email protected]
“velocity set”, a setback for applications. In
this paper we show how the same result remains valid when the convexity assumption is
replaced by a weaker assumption involving merely the mixed constraints. As in [6] we also
consider pointwise set control constraints. Moreover, and again as in [6], a nonsmooth version
of the positive linear independence of the gradients with respect to the control of the function
defining the mixed constraints plays a key role
in validation of our main result. The problem
of interest is (P )

Minimize l(x(0), x(1))




subject to



1 Introduction
ẋ(t) = f (t, x(t), u(t)) a.e. t ∈ [0, 1]
0 > g(t, x(t), u(t)) a.e. t ∈ [0, 1]


Here we focus on necessary conditions for op- 


u(t) ∈ U (t)
a.e. t ∈ [0, 1]

timal control problems with nonsmooth mixed (x(0), x(1)) ∈ C.
constraints. Necessary conditions in the form
of maximum principles for problems with smo- Here the given functions f : [0, 1] × Rn × Rk →
oth mixed constraints have been addressed by Rn , g: [0, 1]×Rn ×Rk → Rm , and multifunction
a number of authors; see for example [9], [13], U : [0, 1] ⇒ Rk describe the system dynamics
[14], [8], [11], to name but a few. However de- and control constraints, while the given set
rivation of necessary conditions covering pro- C ⊂ Rn × Rn and function l: Rn × Rn → R speblems with nonsmooth mixed constraints re- cify the endpoint constraints and costs. Also,
mains a largely unexplored area (an exception a process is a pair (x, u) comprising a funcmay be found in [10] where autonomous pro- tion x ∈ W 1,1 ([0, 1]; Rn ) and a measurable
blems are considered), a surprising fact taking functions u: [0, 1] → Rk . An admissible prointo account the fast development of nonsmo- cess for (P ) is a process satisfying the consoth methods for optimal control since the pu- traints. Here W 1,1 (T ; Rn ) denotes the space
blication of the seminal book [1].
of absolutely continuous functions mapping T
Although a weak maximum principle for op- to Rn . An admissible process (x̄, ū) is a lotimal control problems with nonsmooth inequa- cal minimizer (also known as weak minimility mixed constraints is derived in [6], such re- zer) for (P) if there exists δ 0 > 0 such that
sult holds under a convexity hypothesis on a l(x̄(0), x̄(1)) 6 l(x(0), x(1)) holds for all ad-
Abstract: Necessary conditions of optimality
in the form of a weak maximum principle are
derived for optimal control problems with mixed constraints. Such conditions differ from
previous work since they hold when a certain
convexity assumption is replaced by an “interiority” assumption. Notably the result holds for
problems with possibly nonsmooth mixed constraints and with additional pointwise set control constraints. Essential to all the analysis
is a nonsmooth version of the well known positive linear independence conditions on the mixed constraints.
— 269 —
missible processes (x, u) satisfying the following
condition for almost every t ∈ [0, 1]:
|x(t) − x̄(t)| 6 δ 0 ,
2
|u(t) − ū(t)| 6 δ 0 .
Preliminaries
3
Auxiliary Results:
Case
Scalar
(1) We now focus on a special case of problem
(P ) when the number of inequality mixed constraints is one, i.e., g : [0, 1] × Rn × Rku × Rkv →
R. Define the function
g + (t, x, u) = max {0, g(t, x, u)} .
Rm ,
For g in
inequalities like g 6 0 are interpreted componentwise. We focus on a particular process (x̄, w̄), and write φ̄(t) instead of
φ(t, x̄(t), w̄(t)) for both φ = f and φ = g.
Here and throughout, B represents the closed
unit ball centered at the origin regardless of the
dimension of the underlying space and | · | the
Euclidean norm or the induced matrix norm on
Rp×q . For each t in [0, 1] and some δ > 0, we
define
The following two sets of hypotheses on the
data of this special case of problem (P ), which
make reference to a parameter δ > 0 and a
reference process (x̄, ū), will be of importance:
(H1) For each (x, u) ∈ Rn × Rk , the function t → (f (t, x, u), g(t, x, u)) is Lebesgue
measurable. Also, there exists a function
L ∈ L1 such that both φ = f and φ = g
obey this inequality for almost every t in
[0,
1]:
Tδ (t) = x̄(t) + δB = {y ∈ Rn : |y − x̄(t)| 6 δ} . (2)
|φ(t, x, u) − φ(t, x0 , u0 )| 6 L(t) |(x, u) − (x0 , u0 )|
Likewise we set
for all x, x0 ∈ Tδ (t), u, u0 ∈ Rk .
Uδ (t) = U (t) ∩ (ū(t) + δB).
(3) (H2) The multifunction U has Borel measurable graph. For all δ > 0 sufficiently small,
the set Uδ (t) , as defined in (3), is closed
The Euclidean distance function with respect
k
k
for
almost every t ∈ [0, 1].
to a given set A ⊂ R is a function dA : R → R
defined as dA (y) = inf {|y − x| : x ∈ A} .
(H3) The endpoint constraint set C is closed;
A function h: [0, 1]
→
Rp lies in
the cost function l is locally Lipschitz in a
W 1,1 ([0, 1]; Rp ) if and only if it is absoneighbourhood of (x̄(0), x̄(1)).
lutely continuous; in L1 ([0, 1]; Rp ) iff it is
integrable; and in L∞ ([0, 1]; Rp ) iff it is essen- (H4) Both Kf (t) := |f (t, x̄(t), ū(t))| and
tially bounded. The norm of L1 ([0, 1]; Rp ) is
Kg (t) := |g(t, x̄(t), ū(t))| are integrable on
denoted by k·k1 and the norm of L∞ ([0, 1]; Rp )
[0, 1].
is k·k∞ .
(H5) There exist a constant K1 > 0 and a
We make use of standard concepts from
function h ∈ L∞ ([0, 1]; Rk ), with |h(t)| =
nonsmooth analysis. Let A ⊂ Rk be a closed
1 a.e., such that the following condition
set with x̄ ∈ A. The limiting normal cone to A
is satisfied for almost every t ∈ [0, 1], all
at x̄ is denoted by NA (x̄). Given a lower semi(x, u) ∈ Tδ (t) × Uδ (t) for which g(t, x, u) >
continuous function f : Rk → R ∪ {+∞} and a
0 and all vectors (γ, ψ) ∈ co ∂x,u g(t, x, u):
point x̄ ∈ Rk where f (x̄) < +∞, ∂f (x̄) denotes the limiting subdifferential of f at x̄. When
ψ j · h(t) > K1 .
the function f is Lipschitz continuous near x,
the convex hull of the limiting subdifferential,
Consider additionally the following convexity
co ∂f (x), coincides with the (Clarke) subdiffeassumption:
rential. Properties of Clarke’s subdifferentials
(upper semi-continuity, sum rules, etc.), can be (CC) For almost every t ∈ [0, 1], each
found in [1]. For details on such nonsmooth
of the following sets is convex:
analysis concepts, see [1], [16], [18] (in infinite
V + (t, x) = {(f (t, x, u), g + (t, x, u) + s):
dimensions see also [12]).
u ∈ Uδ (t), s > 0} .
— 270 —
In the context of the special case of problem (P ) under consideration, the unmaximized Hamiltonian is the function H: Rn × Rn ×
Rm × Rk → R defined as H(t, x, p, r, u) :=
p · f (t, x, u) + r · g(t, x, u).
Applying the weak nonsmooth Maximum
Principle given by Theorem 4.1 in [6] to this
special case of (P ) we obtain the proposition
stated below. Clearly the main setback to the
application of this proposition is the restrictive
nature of hypothesis CC.
5
Vector Valued Mixed Constraints
We now focus on the general problem (P ) when
the number of mixed constraints is greater than
1, i.e., we consider g: [0, 1] × Rn Rk → Rm , with
m > 1.
Define the function g + (t, x, u)
=
max {g1 (t, x, u), . . . , gm (t, x, u)} .
Before
establishing our main result we consider two
extra hypotheses:
Proposition 3.1 Let (x̄, ū) be a local minimi- (H5V) There exist a constant K1 > 0 and a
function h ∈ L∞ ([0, 1]; Rk ), with |h(t)| =
zer for problem (P ), with m = 1. Assume H1–
1 a.e., such that the following condition
H5 and CC. Then there exist an absolutely conn
is satisfied for almost every t ∈ [0, 1], all
tinuous function p: [0, 1] → R , integrable funck
m
(x, u) ∈ Tδ (t) × Uδ (t), all j ∈ {1, . . . , m}
tions ξ: [0, 1] → R , and r: [0, 1] → R , and a
for which gj (t, x, u) > 0 and all vectors
scalar λ > 0 such that
(γ j , ψ j ) ∈ co ∂x,u gj (t, x, u):
kpk∞ + λ > 0,
φj · h(t) > K1 .
(−ṗ(t), ξ(t)) ∈co ∂x,u H(t, x̄(t), p(t), r(t), ū(t))
(H6V) For almost every t in [0, 1], we have
ξ(t) ∈ β(t) co ∂dUδ (ū(t)) a.e. t,
{u ∈ Uδ (t) : g + (t, x, u) = 0} 6= ∅, for all
x ∈ Tδ (t).
r(t) · g(t, x̄(t), ū(t)) = 0 and r(t) 6 0 a.e. t,
Observe that H5V is a vector valued version
(p(0), −p(1)) ∈ NC (x̄(0), x̄(1))+ λ∂l(x̄(0), x̄(1)),
of the H5 and that H6V is an adaptation of
where β depends only on δ, L, Kf ∈ the “interiority” hypothesis INT consider in
L1 ([0, 1]; R) and the Lipschitz constant of l.
the subsection . Hypothesis H6V states that
for pairs (x, u) closed to the local minimizer
4 Scalar Case Without Conve- (x̄(t), ū(t)) there exists at least a component of
the function g defining the mixed constraints
xity
that touches the boundary of the admissible
Recall that we are considering (P ) with scalar set. On the other hand, hypothesis H5V is
mixed constraints, that is, we assume that the a nonsmooth version of regularity assumptions
number of inequality mixed constraints is one on the mixed constraints. Assuming smoothness of the function g, H5V coincides with the
(i.e., g : [0, 1] × Rn × Rku × Rkv → R).
well known positive linear independence of the
Consider the following condition:
gradients ∇v gi . In this respect we refer the re(INT) For almost every t in [0, 1], we have ader to [8], [15], [4] and [5]).
{u ∈ Uδ (t) : g(t, x, u) = 0} 6= ∅, for all x ∈
Theorem 5.1 Let (x̄, ū) be a local minimiTδ (t).
zer for problem (P ), with m > 1. Assume
H1–H4, H5V and H6V. Set H(t, x, p, r, u) =
Proposition 4.1 Let (x̄, ū) be a local mini- p·f (t, x, u)+r·g(t, x, u). Then there exist an abmizer for problem (P ) when m = 1 (i.e., solutely continuous function p: [0, 1] → Rn , ing : [0, 1] × Rn × Rk → R). Assume H1–H5 tegrable functions ξ: [0, 1] → Rk , and r: [0, 1] →
and INT. Then there exist an absolutely con- Rm , and a scalar λ > 0 such that
tinuous function p: [0, 1] → Rn , an integrable
kpk∞ + λ > 0,
function ξ: [0, 1] → Rk , an integrable function
(−ṗ(t), ξ(t)) ∈ co ∂x,u H(t, x̄(t), p(t), r(t), ū(t)) a.e. t,
r: [0, 1] → R, and a scalar λ > 0 such that (4)–
ξ(t) ∈ co NUδ (ū(t)) a.e. t,
(4) are satisfied.
The proof of this proposition is presented in
section 6.
— 271 —
r(t) · g(t, x̄(t), ū(t)) = 0 and r(t) 6 0 a.e. t,
(p(0), −p(1)) ∈ NC (x̄(0), x̄(1))+ λ∂l(x̄(0), x̄(1)).
This Theorem results as a corollary of the
Proposition 4.1. The details can be found in
[7] and are omitted.
We would like to add that the above Theorem generalizes the main result in [6] to cover nonconvex problems. In contrast to Theorem 3.1 in [5] this Theorem now covers problems with possibly nonsmooth mixed inequality constraints and, remarkably, pointwise set
control constraints. All the hypotheses under
which Theorem 5.1 holds can be seen as direct adaptation of the hypotheses impose in [5]
with the exception of H6V. Indeed, H6V is an
hypothesis without direct analogous in the literature of necessary conditions for mixed constraints optimal control problems.
6
Proposition 4.1 Proof Outline
The proof breaks in three steps. We first prove
the theorem under the interim hypotheses
3. Suppose that g(t, x, u)
<
0 and
0
g(t, x, u ) > 0. By INT there exists
a u
b ∈ Uδ (t) such that g(t, x, u
b) = 0
and then g + (t, x, u
b) = 0.
Thus
A = αg(t, x, u
b) + (1 − α)g(t, x, u0 ).
Then (4) follows from IH.
Next, an appeal to Proposition 3.1 permit us
to deduce that Proposition 4.1 holds when INT
and IH replace CC. Notice that the fact that g
is a scalar valued function is essential.
Step 2: Removal of IH.
By adjusting δ > 0 we can arrange that (x̄, ū)
is a local minimizer over all admissible processes (x, u) for (P ) such that kx − x̄k∞ 6 δ and
ku − ūk∞ 6 δ. Under our assumptions the set
F (t, x), as defined in IH, is nonempty for each
(t, x) in the set
Ω = {(t, x) ∈ R × Rn : t ∈ [0, 1], x ∈ x̄(t) + δB} .
Write
R := {x ∈ W 1,1 : x(0) ∈ C0 , }
(ẋ(t), 0) ∈ F (t, x(t))}.
(IH) For almost every t ∈ [0, 1], each of the
following sets is convex:
By the Generalized Filippov Selection Theorem
[18, Thm.2.3.13]),x̄ is a minimizer for the proF (t, x) = {(f (t, x, u), g(t, x, u)) : u ∈ Uδ (t)}
blem
for all x ∈ Tδ (t).
Minimize l(x(0), x(1))
n
n
over arcs x in R satisfying kx − x̄kL∞ < δ.
(ECS) C = C0 ×R where C0 ⊂ R is a closed
set.
A straightforward modification of the proof
Step 1: Show that INT and IH imply CC. of the Relaxation Theorem (see, e.g., [18,
Take any u, u0 ∈ Uδ (t), s, s0 > 0 and α ∈ Thm.2.7.2]) implies that any arc x in the set
[0, 1]. Set se = αs + (1 − α)s0 and
Rr := x ∈ W 1,1 : x(0) ∈ C0 ,
A = αg + (t, x, u) + (1 − α)g + (t, x, u0 ).
(ẋ(t), 0) ∈ co F (t, x(t))}
We show that there exists u
e ∈ Uδ (t) such that
+
A + se = g (t, x, u
e) + se,
that is, CC holds. Consider three cases:
which satisfies kx − x̄kL∞ < δ can be approxi(4) mated by an arc y in R satisfying ky − x̄kL∞ <
δ. The continuity of the mapping
x → l(x(0), x(1))
1. If g(t, x, u), g(t, x, u0 ) > 0, then A =
0
αg(t, x, u) + (1 − α)g(t, x, u ) and by IH on a neighbourhood of x̄ implies that x̄ is a
there exists a w
e ∈ Wδ (t) such that (4) minimizer for the optimization problem
holds.
Minimize l(x(0), x(1))
0
2. If g(t, x, u), g(t, x, u )
<
0, then
over arcs x ∈ Rr satisfying kx − x̄kL∞ < δ.
g + (t, x, u) = g(t, x, u0 ) = 0. By INT there
exist u1 , u01 ∈ Uδ (t) such that g(t, x, u1 )
By the Generalized Filippov Selection Theo0
and g(t, x, u1 ) are both zero.
Thus rem and Carathéodory’s Theorem,
A = αg(t, x, u1 ) + (1 − α)g(t, x, u01 ) = 0
{x̄, ȳ ≡ l(x̄(0), x̄(1)), (ū0 , . . . , ūn ) ≡
and, by IH, there exists a u
e ∈ Uδ (t) such
(ū, . . . , ū), (λ0 , λ1 , . . . , λn ) ≡ (1, 0, . . . , 0)}
that (4) holds.
— 272 —
0 0
is a minimizer for the optimization problem (C) li (x, y,
x ,y ) =
max l(x, y) − l(x̄(0), x̄(1)) + ε2i , |x0 − y 0 |} .

Minimize y(1)

Consider


1,1 , y ∈ W 1,1 , and measurable

over
x
∈
W

(



Minimize li (a, b, x(1), b)
functions u0 , . . . , un , λ0 , . . . , λn



(R
)

i

 satisfying
X
subject to (u, a, b) ∈ D.



ẋ(t)
=
λ
(t)f
(t,
x(t),
u
(t)),
a.e.,

i
i


i
Since (ū, x̄(0), x̄(1)) ∈ D, D is nonempty. It
ẏ(t)X
= 0, a.e.


is
a simple matter to check that (D, ∆) is a


0 >
λi (t)g(t, x(t), ui (t)), a.e.


complete metric space on which the functional


i


li : D → R is continuous.
 (λ0 (t), . . . , λn (t)) ∈ Λ,



Notice that li (x̄(0), x̄(1), x̄(1), x̄(1)) = ε2i .

 ui (t) ∈ Uδ (t), i = 0, . . . , n
n a.e.

o

Since li > 0, it follows that the process

 (x(0), x(1), y(0)) ∈ epi e
l + ΨC0 ×Rn ×R .
(ū, x̄(0), x̄(1)) is a “ε2i -minimizer” for (Ri ).
Then we apply Ekeland’s Variational Principle
Here e
l(x, x0 , y) = l(x, x0 ), ΨA is the indicator (Theorem 3.3.1 in [18]). Rewriting the conclufunction of the set A, (ΨA (z) = 0 if x ∈ D and sions in control theoretical terms we obtain a
ΨA (z) = +∞ otherwise),
sequence of perturbed problems to which the
necessary conditions obtained in the previous
i
=
0,
.
.
.
,
n
Λ := {λ00 , . . . , λ0n : λ0i > 0 for
)
step hold. Taking limits we obtain the requin
X
red conclusions.
0
and
λ =1 ,
i
i=0
and (λ0 , . . . , λn ), (u0 , . . . , un ) are regarded as
control variables. This is a problem with a
scalar mixed constraint to which the proposition, as proved in the last step, applies. We
can now write the optimality conditions for this
problem work on them to get the result for the
original problem without convexity. For the details we refer to [].
Step 3: Validation of the result obtained in
Step 2 when hypothesis ECS is removed.
Let D denote the set of pairs (u, a, b) such
that u: [0, 1] → Rk is a measurable function and
(a, b) ∈ C for which there exist absolutely continuous functions (x, y) such that








ẋ(t)
ẏ(t)
0
u(t)
(x(t), y(t)
(x(0), y(0))







=
=
>
∈
∈
∈
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Z
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— 274 —