# Overview on the Pointwise Constrained Liapunov Vectorial Convexity Theorem

Authors: Clara Carlota; Sílvia Chá; António Ornelas
Journal: Conference Papers in Mathematics
 (2013)

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## Abstract

In applications of the Calculus of Variations, Optimal Control and Differential Inclusions, very important real-life problems are nonconvex vectorial and subject to pointwise constraints. The classical Liapunov convexity theorem is a crucial tool allowing researchers to solve nonconvex vectorial problems involving single integrals.
However, the possibility of extending such theorem so as to deal with pointwise constraints has remained an open problem for two decades, in the more realistic case using variable vectorial velocities. We have recently solved it, in the sense of proving necessary conditions and sufficient conditions for solvability of such problem. A quick overview of our results is presented here, the main point being that, somehow, convex constrained nonuniqueness a.e. implies nonconvex constrained existence.


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## Body

## 1. Introduction

Given anyAC (absolutely continuous) function  x¯:[a,b]→ℝn  whose derivatives may be written as convex combinations of  m  velocities  vj(·),  with coefficients  λ¯j(·),
(1)x¯(t)=x¯(a)+∫at∑j=1m‍λ¯j(τ)vj(τ)dτ,
by the 1940 classical theorem of  Liapunov  [1],  on convexity of the range of vector measures, one may partition the interval  [a,b]  into disjoint measurable subsets, having characteristic functions  χEj(·),  such that the new bang-bang function
(2)x^(t):=x¯(a)+∫at∑j=1m‍χEj(τ)vj(τ)dτ
again satisfies the same boundary conditions:
(3)x^(a)=x¯(a),x^(b)=x¯(b).Here we present a simplified overview of the main results of our recent paper  [2],  which has improved this Liapunov theorem by again allowing the fulfilment of the boundary conditions (3)—on the global range of this vector measure—but now under an extra pointwise space constraint:
(4)〈x^(t),ω〉≤〈x¯(t),ω〉∀t∈[a,b],
for any  ω∈ℝn  which one desires to fix once for all, for example ω=(1,0,…,0).Our results have thus generalized  (to any space-dimensionn>1  and any number  m≥2  of velocities  vj(·)  in  (1))  the result of  Amar  and  Cellina  [3, Lemma 3.4]  (valid for  n=1  and  m=2).  They also generalized, to variable vector velocities, the result of  [4]  dealing with constant vector velocities.The many potential applications of our result can be easily guessed by looking at all the applications of the Liapunov convexity theorem to nonconvex differential inclusions, calculus of variations, and optimal control which have been published along the last 3 decades, in particular those due to the last author of this paper and collaborators.Indeed, our motivation to invest such efforts, in trying to solve this difficult problem, has been precisely the need we felt for disposing of the pointwise constrained Liapunov tool in order to unblock our road to solve some important nonconvex vectorial pointwise constrained problems dealing with  (single or multiple)  integrals of the calculus of variations and optimal control.


## 2.  Main Function Spaces and Aim

In order to state the results of the reviewed paper, we start from the space of absolutely continuous vector functions,(5)𝒳:=W1,1([a,b],ℝn),
fix  mintegrable  vector  velocities(6)v1(·),…,vm(·)∈L1([a,b],ℝn),
fix any  x¯(·)  having derivatives in the convex hull  “co”  of these velocities, that is,
(7)x¯(·)∈𝒳co,(8)𝒳co:={x(·)∈𝒳:x′(t)∈co{v1(t),…,vm(t)}a.e.},
and consider the associated  nonconvex  differential  inclusion(9)x′(t)∈{v1(t),…,vm(t)}a.e.,x(a)=x¯(a),x(b)=x¯(b),
and its  convexified,  or  relaxed,  differential  inclusion(10)x′(t)∈co{v1(t),…,vm(t)}a.e.,x(a)=x¯(a),x(b)=x¯(b);
together with their  solution  sets(11)𝒳¯co:={x(·)∈𝒳:x(·)solves(10)},𝒳¯true:={x(·)∈𝒳:x(·)solves(9)}.Fix, moreover, any  “direction”  ω∈ℝn, for example ω=(1,0,…,0),  and consider the corresponding  constrained  solution  set:
(12)𝒳¯-co:={x(·)∈𝒳¯co:〈x(t),ω〉≤〈x¯(t),ω〉∀t∈[a,b]}(constrainedrelaxedsolutions),
where  〈·,·〉  is the inner product in  ℝn.Then the aim of this paper is to establish necessary conditions and sufficient conditions for nonemptiness of the class(13)𝒳¯-true:=𝒳¯true∩𝒳¯-co(constrainedtruesolutions).


## 3. The Geometric Control Problem Viewpoint

The problem we are considering may also be seen as a geometric control problem as follows. Taking, for example,|ω|=1, we are given an AC factor  h¯(·)∈W1,1([a,b]), a hyperplaneH¯(t)orthogonal to ω, namely
(14)H¯(t)∶={S∈ℝn:〈S-ωh¯(t),ω〉=0},t∈[a,b],
moving along the  ω-direction  with  relaxed  velocities(15)h¯′(t)∈co{〈v1(t),ω〉,…,〈vm(t),ω〉}a.e.on[a,b],
and an  AC  point  x¯(t)  moving inside  H¯(t)  with relaxed velocities
(16)x¯′(t)∈co{v1(t),…,vm(t)}a.e.on[a,b];
and we wish to find an AC point  x^(t)  starting from the same initial point,  x^(a)=x¯(a)∈H¯(a),  moving with  true  velocities(17)x^′(t)∈{v1(t),…,vm(t)}a.e.on[a,b],
and reaching exactly the same final endpoint, x^(b)=x¯(b)∈H¯(b),  under the extra constraint of never crossing the  moving  hyperplaneH¯(t).The first result we have obtained says that the final hyperplaneH¯(b)  is always easily attained. Namely, setting
(18)𝒳¯ω:={x(·)∈𝒳:x(a)=x¯(a),〈x(b),ω〉=〈x¯(b),ω〉},𝒳¯ωco∶={x(·)∈𝒳¯ω:x′(t)∈co{v1(t),…,vm(t)}a.e.},𝒳¯ωtrue∶={x(·)∈𝒳¯ω:x′(t)∈{v1(t),…,vm(t)}a.e.},
then our first  (trivial)  vectorial extension  (see  [2, Th. 1.1])  of the scalar pointwise-constrained Liapunov result of  [3]  consists in the following.Theorem 1 (existence of minimal, and maximal, constrained true weak solutions).
(19)
∀
x
¯
(
·
)
∈
𝒳
c
o
∀
ω
∈
ℝ
n
∃
x
ω
min
(
·
)
∈
𝒳
¯
ω
t
r
u
e
∃
x
ω
max
(
·
)
∈
𝒳
¯
ω
t
r
u
e
:
〈
x
ω
min
(
·
)
,
ω
〉
≤
〈
x
(
·
)
,
ω
〉
≤
〈
x
ω
max
(
·
)
,
ω
〉
∀
x
(
·
)
∈
𝒳
¯
ω
c
o
.However, while the final hyperplaneH¯(b)  can always be reached, the exact final target  x¯(b)  inside  H¯(b)  is not always attainable;  so that the real challenge, in this research, has been how to reach, more specifically,  x¯(b).


## 4. Our Generalm×n  Pointwise Constrained Liapunov Convexity Theorem

In order to state the conditions of solvability for the main problem of reaching the exact aimed final endpointx¯(b),  consider a new subspace of  (12)  given by
(20)𝒳¯--co∶={x(·)∈𝒳¯-co:〈x(t),ω〉<〈x¯(t),ω〉a.e.in𝒪x},
where  𝒪x  is an adequate  open  set  which, to simplify exposition, in this overview will be the simplest possible choice:
(21)𝒪x=(a,b)
(see our paper  [2]  for more general definitions of this set).  Then the main result  (see  [2, Theorems 1.2 and 1.4])  of our research is as follows.Theorem 2 (existence of constrained true solutions).
(22)
𝒳
¯
-
-
c
o
≠
∅
⟹
𝒳
¯
-
t
r
u
e
≠
∅
,
∀
m
≥
2
∀
n
≥
1
in
(
6
)
;
or, more precisely, if there exists at least one function in  𝒳¯--co  then  x¯(·)  may be  uniformly  approximated by a  ω-increasing  sequence of  true  solutions:
(23)∃(x^k(·))⊂𝒳¯true:(x^k(·))⟶x¯(·)and(〈x^k(·),ω〉)↗〈x¯(·),ω〉,uniformly.Thus, somehow, convex constrained nonuniqueness a.e. implies nonconvex constrained existence. Moreover, we have proved the following characterization  (see [2, proof of Theorem 1.2]):Theorem 3 (necessary and sufficient condition for existence of constrained true solutions).
One has(24)𝒳¯-true≠∅ifandonlyif∃x~(·)∈𝒳¯-cosuchthatx~′(t)∈{v1(t),…,vm(t)}fora.e.t∈[a,b]∖𝒪,where𝒪:={t∈(a,b):〈x~(t),ω〉<〈x¯(t),ω〉}.


## 5. Nonemptiness of𝒳¯--co  in the Special 2×2 Case

Although (22) provides a nice and simple theoretical sufficient condition for existence of constrained true solutions, in practical applications it is not at all clear, a priori, whether the above class  𝒳¯--co, in (22) and (20)  is, or is not, empty.In this section, we present our first contribution towards clarifying such issue, namely through necessary conditions, and sufficient conditions, to have𝒳¯--co  nonempty and to have  𝒳¯--co  empty. However, for simplicity, we will restrict ourselves to the most basic nontrivial framework:  just  m=2  velocities to be given in  (6),  in space-dimension  n=2;  and with difference between these two velocities having second coordinate writable as a factor  f(t)  times the first one. Thus we assume given
(25)v1(·),v2(·)∈L1([a,b],ℝ2),(v2-v1)(t)=v(t)(1,f(t))a.e.,(26)forsomev(·)≥0a.e.inL1(a,b)andsomef(·)inℱ:={f(·)∈L1(a,b):(f·v)(·)∈L1(a,b)}.While this may sound, at first sight, like a very very special situation, in reality such very basic framework is precisely what we need for our applications, of the constrained Liapunov convexity Theorem2, to the Calculus of Variations involving vector-valued functions in competition towards minimization of nonconvex single or multiple integrals, under state and gradient pointwise constrained Lagrangians.Again for simplicity, let us assume(27)ω=(1,0),
and associate to each  f(·)∈ℱ  the new class of functions
(28)𝒢f∶={g(·)∈W01,1([a,b]):g(·)≢0,(f·g′)(·)∈L1(a,b)}.We begin by presenting a necessary condition  (see [2, Theorem 2.1]).Theorem 4 (necessary condition in2×2 case).
For each givenf(·) and x¯(·),  as in  (25), (26), and (7),  so that
(29)∃λ¯(·)∈L∞(a,b):λ¯(t)∈[0,1]a.e.,x¯′(t)=v1(t)+λ¯(t)v(t)·(1,f(t))a.e.,
it turns out that
(30)if𝒳¯-co∖{x¯(·)}(resp.𝒳¯--co)isnonemptythen∃g¯f(·)∈𝒢f:g¯f(·)≥0(resp.g¯f(·)>0a.e.),∫ab‍f(t)g¯f′(t)dt=0,-[v·(1-λ¯)](t)≤g¯f′(t)≤[v·λ¯](t)a.e.In particular, such necessary condition yields simple counterexamples  (independent ofx¯(·))  illustrating the necessity of including  some  extra hypothesis, for example nonemptiness—as in  (22)—of the set  𝒳¯--co,  in order to guarantee solvability of our pointwise constrained Liapunov problem. Namely,
(31)f(·)a.e.strictlymonotone⟹𝒳¯-co∖{x¯(·)}=∅(see(12));
or, as an alternative,
(32)f(·)a.e.monotoneand∫ab‍|f′(t)|dt>0⟹𝒳¯--co=∅(see(20))
(or, more precisely,  ∃  monotone  fˇ(·)=f(·)  a.e.  and  ∫ab‍|fˇ′(t)|dt>0).On the other hand, in order to reach a sufficient condition guaranteeing𝒳¯--co≠∅,  let us associate to each  f(·)  as in  (25) and (26)  and to each  d∈{1,2,…}  a
(33)d×d-symmetricmatrixMfd
whose entries
(34)Mfij:=(-1)k(k+1)!i!j!×[(k+2)f(-k-2)(b)-(b-a)f(-k-1)(b)],
for  k:=i+j  and i,j∈{0,1,…,d-1},  are linear combinations of pairs of iterated indefinite integrals of  f(·):(35)f(0)(t):=f(t)f(-k-1)(t):=∫at‍f(-k)(τ)dτfork=0,1,2,….Then, given any reference relaxed solutionx¯(·) hence a corresponding  λ¯(·),  as in  (7) and (29),  we finally present our main result on the special  2×2  case  (see  [2, Theorem 2.5]).Theorem 5 (Sufficient Condition in2×2 case).
Under  (25), (26), (27), and (29),  assume the matrix  Mfd  has,
(36)forsomei∈{1,2,…},Mfii·Mf00≤0(or,moregenerally,Mfii·Mf00≤(Mf0i)2)
or, still more generally,  Mfd  has
(37)eigenvaluesβ-,β+withβ-·β+≤0,
and the following inequalities hold true:  (see  (26) and (29))
(38)-essinf[v·(1-λ¯)]([a,b])<0<essinf[v·λ¯]([a,b]).
Then(39)∃x~(·)∈𝒳¯--cohavingx~′(·):=x¯′(·)-g′(·)·(1,f(·)),
with a  2d-degree polynomial  g(·)>0  a.e. having  g(a)=0=g(b)  and satisfying  (30).


## 6. Open Problems and Challenges

While we have thus obtained nice and useful research results, the truth is that each conquest always raises new and interesting questions whose answer is unknown, hence opening new challenges for future researches.In particular, we are already working on extending the results here presented, namely to optimal control problems steered by linear differential inclusions of first order, which poses some nice challenges,  and of higher order, posing still harder challenges. Another related question is the application of these results to optimal control problems involving minimization of pointwise-constrained multiple integrals, which was in reality our original motivation.We would welcome brave Masters or Ph.D. students interested in following and contributing to our researches in this field.



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