MINISTRY OF EDUCATION AND TRAININGHANOI PEDAGOGICAL UNIVERSITY 2PHAN TRONG TIEN β-VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS AND APPLICATIONS TO A CLASS OF OPTIMAL CONTROL PROBLEMS
Trang 1MINISTRY OF EDUCATION AND TRAININGHANOI PEDAGOGICAL UNIVERSITY 2
PHAN TRONG TIEN
β-VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS
AND APPLICATIONS TO
A CLASS OF OPTIMAL
CONTROL PROBLEMS
Major: AnalysisCode: 9 46 01 02
Summary of Doctoral Thesis in Mathematics
Hanoi-2020
Trang 2This dissertation is completed at:
Hanoi Pedagogical University 2
Supervisor: Dr Tran Van Bang
Assoc Prof Dr Ha Tien Ngoan
First referee:
Second referee:
Third referee:
The thesis shall be defended at the University level Thesis As-sessment Council at Hanoi Pedagogical University 2 on
2019 at oclock
The dissertation is publicly available at:
- The National Library of Vietnam
- The Library of Hanoi Pedagogical University 2
Trang 3First-order Hamilton-Jacobi equations (HJEs) comprise an portant class of nonlinear partial differential equations (PDEs) withmany applications Typical examples can be found in mechanics,optimal control theory, etc Specifically, this class includes dynamicprogramming equations arising in deterministic optimal controls,which are known as Hamilton-Jacobi-Bellman equations In gen-eral, these nonlinear equations do not have classical solutions As aresult, it is necessary to study weak solutions and a viscosity solution
im-is such a week solution
The theory of viscosity solutions for partial differential equationsappeared in 1980s In particular, in the paper by M G Crandalland P L Lions (1983), the authors introduced the viscosity solution
as a generalized solution of partial differential equations Instead ofrequiring that the solution u satisfies the given equation almost ev-erywhere, it is sufficient for u to be a continuous functions satisfying
a pair of inequalities via sufficiently smooth test functions, or viasubdifferential and superdifferential
The viscosity solution is an effective device to study nonlinearHamilton-Jacobi equations We emphasize that a viscosity solution
is a weak solution since it is only continuous and its derivative isdefined through test functions and the extremal principle However,
it has been proved that viscosity solution can be defined by ferential, superdifferential, which are called semiderivatives It leads
subdif-to a tight connection between the theory of viscosity solution andnonsmooth analysis which includes subdifferential theory
Since 1993, the smooth variational principle, which was proved
by Deville, has been widely employed as an important tool to lish the uniqueness of β-viscosity solution, in the class of continuousand bounded functions, of Hamilton-Jacobi equations of the form
estab-u + F (Destab-u) = f , where F is estab-uniformly continestab-uoestab-us on Xβ∗ and f isuniformly continuous and bounded on X
Optimal control problems were introduced in 1950s It is wellknown that they have many applications in Mathematics, Physics,and application areas By the dynamic programming principle, thevalue function of an optimal control problem is a solution to an
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Trang 4associated partial differential equation Unfortunately, since valuefunctions might not be differentiable, several approaches have beenintroduced to study them The viscosity solution again is an effec-tive approach to investigate optimal control theory To the best ofour knowledge, treating optimal control problems by viscosity solu-tions via subdiffrential is scared especially if the value function isunbounded.
Recently, an increasing literature has been devoted to to thestudy of HamiltonJacobi equation on junctions and networks Theauthors established properties of the value function, the comparisonprinciple for optimal control problems with bounded running cost l.Although many important results have been obtained, it seems thatthe assumptions in the recent work are quite strict
We focus on β-subdifferential, the uniqueness of β-viscosity lution for Hamilton-Jacobi equations of the forms u + H(x, Du) = 0and u + H(x, u, Du) = 0, the existence and stability of β-viscositysolution Moreover, there are many applications of β-viscosity so-lutions for optimal control problems Motivated by that fact, weare also interested in finding necessary and sufficient conditions foroptimal control problems in infinite dimensional spaces The newapproach of viscosity solution on junctions is another topic of ourinterest Based on the known model of classical viscosity solution,the uniqueness and applications of viscosity solutions for optimalcontrol problems on junctions are promising topics
so-In addition to so-Introduction, Conclusion, and References List, thedissertation consists of four chapters
In Chapter 1, we present the notion of β-viscosity solution and itsproperties, and several results on the smooth variational principle
in Chapter 2, We prove the uniqueness of β-viscosity solution forHamilton-Jacobi equations of the general form u + H(x, u, Du) = 0
in Banach spaces The stability and existence of the solution of suchequations are also investigated
In Chapter 3, we show that the value function of a certain timal control problem is a β-viscosity solution of the associatedHamilton-Jacobi equation The feedback controls and also sufficientconditions for optimality are also studied in this chapter
op-In Chapter 4, we present the notion of junctions, assumptions
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Trang 5and optimal control problems on junctions Several properties of thevalue function such as the continuity on G, the local Lipschitz at O
on each Ji, estimates of the value function at O through Hamilton
We prove that the value function of an optimal control problem onjunctions is a viscosity solution of the associated Hamilton-Jacobiequation We also apply our results in such optimal control problem
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Trang 6Chapter 1
β β-SUBDERIVATIVE
In this chapter, we present β-viscosity subdifferential on Banachspace X and prove the smooth β-variational principle which will beused to establish the uniquess of β-viscosity solution
1.1 β-differentiable
Definition 1.1.1 A borno β on X is a family of closed, bounded,and centrally symmetric subsets of X satisfying the following threeconditions:
1) X = S
B∈β
B;
2) β is closed under scalar multiplication,
3) the union of any two elements in β is contained in some ment of β
ele-By Theorem 27 in [Hoang Tuy, 2005], a borno β in Definition1.1.1 defines on X∗ a locally convex Hausdorff topology τβ Thespace X∗ with this topology τβ is denoted by Xβ∗ A local base ofthe origin 0 in Xβ∗ is the collection of sets of the form
{f : |f (x)| < ε, ∀x ∈ M },where > 0 is arbitrry and M ∈ β
Then, the sequence (fm) ⊂ X∗, converges to f ∈ X∗with respect
to τβ if and only if for any M ∈ β and any ε > 0, there exists n0 ∈ Nsuch that |fm(x) − f (x)| < ε for all m ≥ n0 and x ∈ M ; that is, fmconverges uniformly to f on M Hence τβis also called the uniformlyconvergent topology on elements of β
Example 1.1.2 It is easy to verify the following facts 1) Thefamily F of all closed, bounded, and centrally symmetric subsets of
X is a borno on X, which is called Fr´echet borno
2) The family H of all compact, centrally symmetric subsets of X
is a borno on X called Hadamard borno
3) The family W H of all weakly compact, closed, and centrally metric subsets of X is a borno on X called weak Hadamard borno
sym-4
Trang 74) The family G of all finite, centrally symmetric subsets of X isalso a borno on X called Gˆateaux borno.
Remark 1.1.3 If β borno is F (Fr´echet), H (Hadamard), W H(Hadamard weak) or G (Gˆateaux), then we have Fr´echet topology,Hadamard topology, Hadamard weak topology and Gˆateaux topology
on the dual space X∗, respectively Thus, F -topology is the strongesttopology and G topology is the weakest topology among β-topologies
f (x0+ th) − f (x0) − hp, thi
uniformly in h ∈ V for every V ∈ β
We say that the function f is β-smooth at x0 if there exists
a neighborhood U of x0 such that f is β-differentiable on U and
∇βf : U → Xβ∗ is continuous
1.2 β-viscosity subdifferential
Definition 1.2.1 Let f : X → R be a lower semicontinuous tion and f (x) < +∞ We say that f is β-viscosity subdifferentiableand x∗ is a β-viscosity subderivative of f at x if there exists a lo-cal Lipschitzian function g : X → R such that g is β-smooth at x,
func-∇βg(x) = x∗ and f − g attains a local minimum at x We denotethe set of all β-subderivatives of f at x by D−βf (x), which is calledβ-viscosity subdifferential of f at x
Let f : X → R be an upper semicontinuous function and f (x) >
−∞ We say that f is β-viscosity superdifferentiable and x∗ is a viscosity superderivative of f at x if there exists a local Lipschitzianfunction g : X → R such that g is β-smooth at x, ∇βg(x) = x∗and f − g attains a local maximum at x We denote the set of allβ-superderivatives of f at x by D+βf (x), which is called β-viscositysuperdifferential of f at x
β-Theorem 1.2.2 1) If β1 ⊂ β2 then Dβ−
2f (x) ⊂ Dβ−
1f (x); inparticular, D−Ff (x) ⊂ Dβ−f (x) ⊂ DG−f (x) for every borno β
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Trang 82) If f is continuous, f (x) is finite and D−βf (x), D+βf (x) are twononempty sets, then f is β-differentiable at x.
3) If β1 ⊂ β2 and f is β1-differentiable at x and f is β2-viscositysubdifferentiable at x, then Dβ−
Theorem 1.2.4 If f is a convex function defined on the convex set
C and x ∈ C, then for every borno β we have
D−βf (x) = DG−f (x) = ∂f (x)
Next, we denote
Dβ(X) = {g : X → R |g is bounded, Lipschitzian, and
β-differentiable on X},kgk∞= sup{|g(x)| : x ∈ X}, k∇βgk∞= sup{k∇βg(x)k : x ∈ X}and
D∗β(X) = {g ∈ Dβ(X)| ∇βg : X → Xβ∗ is continuous}.The following hypotheses will be used in the derivation of ourresults
(Hβ) There exists a bump function b such that b ∈ Dβ(X); and(Hβ∗) There exists a bump function b (i.e its support is nonemptyand bounded) such that b ∈ Dβ∗(X)
Proposition 1.2.5 The hypotheses (Hβ) and (Hβ∗) are fulfilled ifthe Banach space X has a β-smooth norm
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Trang 9Proposition 1.2.6 Let X be a Banach space satisfying (Hβ) (resp.(Hβ∗)) and E a closed subset of X Then, for a lower semicontinuousbounded from below function f on E and any ε ∈ (0, 1), there exist
a g ∈ Dβ(X) (resp g ∈ D∗β(X)) and an x0∈ E such that:
(a) f + g attains its minimum at x0
β-lim
η→0inf{
NXn=1
n=1
fn(xn) < inf
x∈X
NXn=1
fn(x) + ε;
(iii)
NXn=1
Trang 10(i) diam(x1, · · · , xN) max(1, kx∗1k, · · · , kx∗Nk) < ε;
(ii)
NX
n=1
fn(xn) < inf
x∈Ω
NXn=1
fn(x) + ε;
(iii)
NXn=1
x∗n < ε
Conclusion
In Chapter 1, we have focused on the following:
1) We have given some remarks about the β-differentiable, the tionship between the β-differentiable when the borno β is implicit
rela-We have also provided several remarks on common subdifferentialsand their relations In addition, we have pointed out certain cases inwhich the different functions have the same set of subdifferentials.2) We have proved the addition rules of m sums of β-subdifferential
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Trang 110 on a set Ω ⊂ X the doubling of variables technique Our resultsare established on a Banach space X with a β-smooth norm or anorm being equivalent to a β-smooth norm without using the Radon-Nikodym assumption We also show the existence, uniqueness, andthe stability of the solution The results in this chapter are based
on the paper [1] in the list of scientific publications related to thisdissertation In this dissertation, the solution existence of Dirichletproblem is proved under an additional assumption that there areequal subsolution and a supersolution on the boundary (comparedwith the existence result in [1]) In addition, we prove another re-sult on the existence of a solution for Hamilton-Jacobi equations(Theorem 2.2.2)
2.1 The uniqueness of β-viscosity solutions
Let X be a real Banach space with a β-smooth norm | · |, Ω ⊂ X
an open subset We study the existence, uniqueness and stability ofβ-viscosity solutions for the following HJEs
u + H(x, u, Du) = 0 in Ω, (2.1)subject to the boundary condition (in the case Ω 6= X)
Here, u : Ω → R and ϕ : ∂Ω → R and H : Ω × R ×Xβ∗ → Rare merely continuous in general, where Xβ∗ is the dual space of theBanach space X, and equipped with topology τβ (see Definition ??).2.1.1 β-viscosity solutions
Definition 2.1.1 A function u : Ω → R is said to be
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Trang 12(i) a β-viscosity subsolution of (2.1) if u is upper semicontinuousand for any x ∈ Ω, x∗∈ Dβ+u(x), F (x, u(x), x∗) ≤ 0;
(ii) a β-viscosity supersolution of ( (2.1)) if u is lower uous and for any x ∈ Ω, x∗ ∈ D−βu(x), F (x, u(x), x∗) ≥ 0;(iii) a β-viscosity solution of ( (2.1)) if u is simultaneously a β-viscosity subsolution and a β-viscosity supersolution
semicontin-For convenience, hereafter, we will use the phrases “β-viscositysolution of H ≤ 0“ and “β-viscosity subsolution of H = 0” in-terchangeably Similarly for the phrases “β-viscosity solution of
H ≥ 0“ and “β-viscosity supersolution of H = 0”
Definition 2.1.2 A function u : Ω → R is said to be a β-viscositysubsolution (resp supersolution, solution) of the problem (2.1)-(2.2)iff u is a β-viscosity subsolution (resp supersolution, solution) ofEquation (2.1) and u ≤ ϕ (resp u ≥ ϕ, u = ϕ) on ∂Ω
Next, we make the following assumptions on the function H.(H0) There exists a continuous function wR : Xβ∗ → R for each
R > 0, satisfying
|H(x, r, p) − H(x, r, q)| ≤ wR(p − q)whenever x ∈ X, p, q ∈ X∗ and r ∈ R satisfy |x|, |q|, |p| ≤ R.(H1) For each (x, p) ∈ X × X∗, r 7→ H(x, r, p) is nondecreasing.(H1)∗ For each (x, p) ∈ X × X∗, r 7→ H(x, r, p) is Lipschitz continu-ous with constant L < 1
(H2) There is a local modulus σH such that
H(x, r, p) − H(x, r, p + q) ≤ σH(|q|, |p| + |q|)for all r ∈ R, x ∈ Ω and p, q ∈ X∗
(H3) There is a modulus mH such that
H(y, r, λ(∇β| · |2)(x − y))−H(x, r, λ(∇β| · |2)(x − y))
≤ mH(λ|x − y|2+ |x − y|) (2.3)for all x, y ∈ Ω with x 6= y, r ∈ R and λ ≥ 0
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Trang 132.1.2 Bounded solutions
Theorem 2.1.3 Let X be a Banach space with an equivalent smooth norm Suppose that F (x, u, Du) = u + H(x, Du) with H :
β-X × β-Xβ∗ → R satisfy the following assumption:
(B) for any x, y ∈ X and x∗, y∗∈ Xβ∗,
|H(x, x∗)−H(y, y∗)| ≤ w(x−y, x∗−y∗)+K max(kx∗k, ky∗k)kx−yk,where K is a constant and w : X × Xβ∗ → R is continuous functionwith w(0, 0) = 0
Let u, v be two bounded functions such that u is upper ous and v is lower semicontinuous If u is a β-viscosity subsolutionand v is a β-viscosity supersolution of equations F (x, u, Du) = 0then u ≤ v
semicontinu-Corollary 2.1.4 Under the assumptions of Theorem 2.1.3, β-viscositysolutions continuous and bounded of equations u + H(x, Du) = 0 isunique
Theorem 2.1.5 Let X be a Banach space with an equivalent smooth norm Ω ⊂ X an open subset
β-Suppose F (x, u, Du) = u + H(x, Du) with H : X × Xβ∗ → R satisfythe following assumption:
(C) for any x, y ∈ X and x∗, y∗ ∈ Xβ∗,
|H(x, x∗)−H(y, y∗)| ≤ w(x−y, x∗−y∗)+K max(kx∗k, ky∗k)kx−yk,wwhere K is a constant and w : X × Xβ∗→ R is continuous functionwith w(0, 0) = 0
Let u, v be two uniformly continuous bounded on Ω If u is a viscosity subsolution and v is a β-viscosity supersolution of equations
β-F (x, u, Du) = 0 and u ≤ v on ∂Ω then u ≤ v on Ω
Corollary 2.1.6 Under the assumptions of Theorem 2.1.5, u, v betwo uniformly continuous bounded on Ω such that u = v on ∂Ω If
u, v be two β-viscosity solution F (x, u, Du) = 0 then u = v on Ω.2.1.3 Unbounded solutions
Based on the preparation in the preceding sections, now wepresent the main results on the uniqueness of the β-viscosity of (2.1)
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