Hamilton–Jacobi–Bellman Equations on Multi-domains

Abstract

A system of Hamilton–Jacobi (HJ) equations on a partition of $\mathbb {R}^{d}$ is considered, and a uniqueness and existence result of viscosity solution is analyzed. While the notion of viscosity solution is by now well known, the question of uniqueness of solution, when the Hamiltonian is discontinuous, remains an important issue. A uniqueness result has been derived for a class of problems, where the behavior of the solution, in the region of discontinuity of the Hamiltonian, is assumed to be irrelevant and can be ignored (see (Camilli, Siconolfi in Adv. Differ. Equ. 8(6):733–768, 2003)). Here, we provide a new uniqueness result for a more general class of Hamilton–Jacobi equations.

The work has been partially supported by the European Union under the 7th Framework Programme “FP7-PEOPLE-2010-ITN”, GA number 264735-SADCO.

Appendices

Appendix A

Proof of Lemma 2.2

Note that although G is only usc on $\mathbb {R}^{d}$, G is Lipschitz continuous on Γ _j since G is the convexification of a finite group of Lipschitz continuous multifunctions on Γ _j. For any x∈Γ _j, there exists α>0 and a diffeomorphism $g\in C^{1,1}(\mathbb {R}^{d})$ such that

$$\overline{B(x,\alpha)\cap\varGamma_j}=\bigl\{x:g(x)=0\bigr\}\quad \mbox{and}\quad \nabla g(y)\neq0,\quad \forall y\in\overline{B(x,\alpha)}. $$

We can take g as the signed distance function to Γ _j for instance. Then there exists β>0 such that

$$\bigl\|\nabla g(y)\bigr\|\geq\beta,\quad \forall y\in B(x,\alpha)\cap\varGamma_j. $$

For any $w\in G(x)\cap \mathcal {T}_{\varGamma_{j}}(x)$, by the Lipschitz continuity of G there exists v∈G(y) such that

$$\|w-v\|\leq L_G\|x-y\|, $$

where L _G is the Lipschitz constant of G(⋅). Since $w\in \mathcal {T}_{\varGamma_{j}}(x)$, we have

$$w\cdot\nabla g(x)=0. $$

Then

$$\bigl\|v\cdot\nabla g(x)\bigr\|=\bigl\|(v-w)\cdot\nabla g(x)\bigr\|\leq L_G\|\nabla g\| \|x-y\|. $$

Thus,

$$ \begin{aligned} \bigl\|v\cdot\nabla g(y)\bigr\|&\leq\bigl\|v\cdot\nabla g(x)\bigr\|+\bigl\| v\cdot\bigl( \nabla g(y)-\nabla g(x)\bigr)\bigr\| \\ &\leq\bigl(L_G\|\nabla g\|+\|G\|L_g' \bigr)\|x-y\|, \end{aligned} $$

where $L_{g}'$ is the Lipschitz constant of ∇g(⋅). We consider the following three cases:

• If v⋅∇g(y)=0, then $v\in \mathcal {T}_{\varGamma_{j}}(y)$ and we deduce that

  $$w\in G(y)\cap \mathcal {T}_{\varGamma_j}(y)+L_G\|x-y\|B(0,1). $$

• If v⋅∇g(y):=−γ<0, let p:=δ∇g(y)/∥∇g(y)∥, then by (H4)(iv),

  $$p\in G(y)\quad \mbox{and}\quad p\cdot\nabla g(y):=\tilde{\beta}\geq\delta\beta>0. $$

We set

$$q:=\frac{\tilde{\beta}}{\tilde{\beta}+\gamma}v +\frac{\gamma}{\tilde {\beta}+\gamma}p, $$

then q⋅∇g(y)=0, i.e. $q\in \mathcal {T}_{\varGamma_{j}}(y)$. And since G(y) is convex, we have q∈G(y). Then we obtain

$$ \begin{aligned} \|w-q\|&\leq\|w-v\|+\|v-q\| \\ &\leq L_G\|x-y\|+\frac{\gamma}{\tilde{\beta}+\gamma}\|v-p\| \\ &\leq\biggl(L_G+\frac{L_G\|\nabla g\|+\|G\|L_g'}{\delta\beta}2\|G\| \biggr)\|x-y\|, \end{aligned} $$

where we deduce that

$$ w\in G(y)\cap \mathcal {T}_{\varGamma_j}(y)+ L\|x-y\|B(0,1), $$

(A.1)

with $L:=L_{G}+2\|G\|(L_{G}\|\nabla g\|+\|G\|L_{g}')/\delta\beta$.

If v⋅∇g(y)>0, then by the same argument taking p=−δ∇g(y)/∥∇g(y)∥, (A.1) holds true as well.

Finally, (A.1) implies the local Lipschitz continuity of $G(\cdot)\cap \mathcal {T}_{\varGamma_{j}}(\cdot)$ on Γ _j with the local constant L. □

Proof of Lemma 3.6

For k=1,…,m+ℓ, we set

For each $k\in\mathbb{K}(x)$, we have $x\in \mathcal {M}_{k}$. Up to a subsequence, there exists 0≤λ _k≤1 and $p_{k}\in \mathbb {R}^{d}$ so that

$$\frac{\mu^n_k}{t_n}\rightarrow \lambda_k,\qquad \sum _{k\in\mathbb{K}(x)}\lambda_k=1,\qquad \frac{1}{\mu^n_k}\int _{J^n_k}\dot{y}(s)ds\rightarrow p_k $$

as n→+∞. By Proposition 3.4 and the Lipschitz continuity of $F^{\mathit {new}}_{k}$, we have

$$ \begin{aligned} p_k&=\lim_{n\rightarrow \infty}\frac{1}{\mu^n_k} \int_{J^n_k}\dot{y}(s)ds \\ &\in\lim_{n\rightarrow \infty}\frac{1}{\mu^n_k}\int_{J^n_k} F^{\mathit {new}}_k\bigl(y(s)\bigr)ds \\ &\subseteq\lim_{n\rightarrow \infty} \biggl[\frac{1}{\mu^n_k}\int _{J^n_k}F^{\mathit {new}}_k(x)ds+ \frac{1}{\mu^n_k}\int _{J^n_k}L_k\bigl\|y(s)-x\bigr\|B(0,1)ds \biggr] \\ &\subseteq\lim_{n\rightarrow \infty} \biggl[F^{\mathit {new}}_k(x)+L_k \|F\| \biggl[\frac{1}{\mu^n_k}\int_{J^n_k}sds \biggr]B(0,1) \biggr]=F^{\mathit {new}}_k(x). \end{aligned} $$

We then have

$$ \begin{aligned} p&=\lim_{n\rightarrow \infty}\frac{y(t_n)-x}{t_n} =\lim _{n\rightarrow \infty}\frac{1}{t_n}\int^{t_n}_0 \dot{y}(s)ds \\ &=\sum_{k\in\mathbb{K}(x)}\lim_{n\rightarrow \infty} \frac{\mu^n_k}{t_n} \biggl[\frac{1}{\mu^n_k}\int_{J^n_k} \dot{y}(s)ds \biggr] \\ &=\sum_{k\in\mathbb{K}(x)}\lambda_k p_k \in\sum_{k\in\mathbb{K}(x)}\lambda_k F^{\mathit {new}}_k(x) \subseteq \text {co }\bigcup_{k\in\mathbb{K}(x)} F^{\mathit {new}}_k(x). \end{aligned} $$

Now set $\mathcal {M}:=\cup_{k\in\mathbb{K}(x)}\mathcal {M}_{k}$, and since $y(t_{n})\in \mathcal {M}$ for all large n, we have $p\in \mathcal {T}_{\overline{\mathcal {M}}}(x)$. Then we obtain

$$p\in\biggl(\text {co }\bigcup_{k\in\mathbb{K}(x)}F^{\mathit {new}}_k(x) \biggr) \cap \mathcal {T}_{\overline{\mathcal {M}}}(x). $$

The fact that $F^{\mathit {new}}_{k}(z)\subseteq \mathcal {T}_{\mathcal {M}_{k}}(z)$ whenever $z\in \mathcal {M}_{k}$ implies

$$F^{\mathit {new}}_k(x)\cap \mathcal {T}_{\overline{\mathcal {M}}}(x)=F^{\mathit {new}}_k(x) \cap \mathcal {T}_{\overline{\mathcal {M}}_k}(x) $$

whenever $x\in\overline{\mathcal {M}}_{k}$. Hence

 □

Appendix B

We review the background in nonsmooth analysis required in our analysis. A closed set $\mathcal {C}\subseteq \mathbb {R}^{d}$ is called proximally smooth of radius δ>0 provided the distance function $d_{\mathcal {C}}(x):=\inf_{c\in \mathcal {C}}\|c-x\|$ is differentiable on the open neighborhood $\mathcal {C}+\delta B(0,1)$ of $\mathcal {C}$. For any $c\in \mathcal {C}$, we denote the Clarke normal cone by $\mathcal{N}_{\mathcal {C}}(c)$. Recall the tangent cone $\mathcal {T}_{\mathcal {C}}(c)$ at $c\in \mathcal {C}$ is defined as

$$ \mathcal {T}_{\mathcal {C}}(c)= \biggl\{v:\liminf_{t\rightarrow 0^-} \frac{d_{\mathcal {C}}(c+tv)}{t}=0 \biggr\}, $$

and in the case of $\mathcal {C}$ proximally smooth, equals the Clarke tangent cone as the negative polar of $\mathcal{N}_{\mathcal {C}}(c)$:

$$v\in \mathcal {T}_{\mathcal {C}}(c)\quad\Longleftrightarrow\quad\langle\zeta,v\rangle \leq0\quad\forall\zeta\in\mathcal{N}_{\mathcal {C}}(c). $$

If $\mathcal {M}$ is an embedded C ² manifold, $\mathcal {C}:=\overline{\mathcal {M}}$, and $c\in \mathcal {M}$, then $\mathcal {T}_{\mathcal {C}}(c)$ agrees with the usual tangent space $\mathcal {T}_{\mathcal {M}}(c)$ to $\mathcal {M}$ at c from differential geometry (see [9, Proposition 1.9]). If in addition $\overline{\mathcal {M}}$ is proximally smooth, then for each $x\in\overline{\mathcal {M}}$, the tangent cone $\mathcal {T}_{\overline{\mathcal {M}}}(x)$ is closed and convex, and thus has a relative interior denoted by $\text {\emph {r-int} }\mathcal {T}_{\overline{\mathcal {M}}}(x)$. Its relative boundary is defined as $\text {\emph {r-bdry} }\mathcal {T}_{\overline{\mathcal {M}}}(x):=\mathcal {T}_{\overline{\mathcal {M}}}(x)\backslash \text {\emph {r-int} }\mathcal {T}_{\overline{\mathcal {M}}}(x)$.

Another key assumption on the multi-domains is each domain being relatively wedged. A set $\mathcal {C}\subseteq \mathbb {R}^{N}$ is wedged (see [9, p.166]) if at every $x\in \text {bdry }\mathcal {C}$, $\mathit{int}\mathcal {T}_{\overline{\mathcal {C}}}\not=\emptyset$. If $\mathcal {C}=\overline{\mathcal {M}}$ is the closure of an embedded manifold $\mathcal {M}$, then $\mathcal {C}$ relatively wedged means the dimension of $\text {\emph {r-int} }\mathcal {T}_{\overline{\mathcal {M}}_{k}}(x)$ is equal to d _k.

The following result is [5, Lemma 3.1] and is the key geometrical ingredient that permits the construction of boundary trajectories of (DI).

Lemma B.1

If $x\in\overline{\mathcal {M}_{k}}\backslash \mathcal {M}_{k}$and $v\in \text {\emph {r-bdry} }\mathcal {T}_{\overline{\mathcal {M}_{k}}}(x)$, then there exists an indexjfor which $\mathcal {M}_{j}\subseteq \overline{\mathcal {M}_{j}}$, $x\in\overline{\mathcal {M}}_{j}$, and $v\in \mathcal {T}_{\overline{\mathcal {M}_{k}}}(x)$. Of course in this case, one hasd _j<d _k.

Proof

See [5, Lemma 3.1]. □

We finally give a sketch of the proof for Lemma 3.8.

Proof

A key fact is that for any $p\in G(x)\cap \text {\emph {r-int} }\mathcal {T}_{\overline{\mathcal {M}}_{k}}(x)$, there exist τ>0 and a C ¹ trajectory $y(\cdot):[t,\tau]\to{ \mathcal {M}}_{k}\cup\{x\}$ so that

$$ y(t)=x \quad\mbox{and}\quad\dot{y}(t)=p. $$

(B.1)

Let k be such that $x\in\overline{\mathcal {M}}_{k}$ and $p\in G(x)\cap \mathcal {T}_{\overline{\mathcal {M}}_{k}}(x)$. If $p\in \text {\emph {r-int} }\mathcal {T}_{\overline{\mathcal {M}}_{k}}(x)$, then the result follows by the key fact (B.1). If $p\notin \text {\emph {r-int} }\mathcal {T}_{\overline{\mathcal {M}}_{k}}(x)$, then $p\in \text {\emph {r-bdry} }\mathcal {T}_{\overline{\mathcal {M}}_{k}}(x)$ and hence by Lemma B.1, there exists another subdomain $\mathcal {M}_{j}\subseteq \overline{\mathcal {M}}_{k}$ with $x\in\overline{\mathcal {M}}_{j}$ and $p\in \mathcal {T}_{\overline{\mathcal {M}}_{j}}(x)$. If $p\in \text {\emph {r-int} }\mathcal {T}_{\overline{\mathcal {M}}_{j}}(x)$, then the result follows from (B.1), otherwise the argument just given can be repeated with k replaced by j. The process must eventually terminate since the dimension is decreasing at each step. □