Stochastic Optimal Control of a Doubly Nonlinear PDE Driven by Multiplicative Lévy Noise

Abstract

In this work, we study an initial value control problem for a doubly nonlinear PDE perturbed by multiplicative Lévy noise. We first establish wellposedness of a weak martingale solution. Monotonicity arguments have been exploited in the proofs. A path-wise uniqueness of weak martingale solutions is settled via the standard $L^1$-method. We formulate the associated control problem, and establish existence of a weak optimal solution of the underlying problem via variational approach.

5 Appendix

In this Appendix, we wish to prove Theorem 3.5. To do so, we start with some useful definition. For any separable and complete metric space $({\mathbb {S}}, \rho )$, let us recall the notion of modulus of a function in the space ${\mathbb {D}}(0,T;{\mathbb {S}})$ endowed with the Skorokhod topology.

Definition 5.1

Let $\delta >0$ be given and $u\in {\mathbb {D}}(0,T;{\mathbb {S}})$. Let $\Pi _\delta$ be the set of all increasing sequence of partitions ${\bar{p}}=\{0=t_0<t_1<t_2<\ldots <t_n=T\}$ of [0, T] with the property $t_{i+1}-t_i \ge \delta$ for $i=0,1,\ldots , n-1$. A modulus of u in the space ${\mathbb {D}}(0,T;{\mathbb {S}})$, denoted by $w_{[0,T];{\mathbb {S}}}(u,\delta )$, is defined by

$$\begin{aligned} w_{[0,T];{\mathbb {S}}}(u,\delta )= \inf _{\Pi _\delta } \max _{t_i \in {\bar{p}}} \sup _{t_i \le s<t <t_{i+1}\le T} \rho (u(t),u(s))\,. \end{aligned}$$

Analogous to the Arzela-Ascoli theorem for the space of continuous functions, we have the following criterion for relative compactness of a subset of the space ${\mathbb {D}}(0,T;{\mathbb {S}})$, whose proof can be found in [16, 21].

Theorem 5.1

A set ${\mathcal {A}} \subset {\mathbb {D}}(0,T;{\mathbb {S}})$ has a compact closure if and only if it satisfies the following two conditions:

• (i) there exists a dense subset ${\bar{I}}$ of [0, T] such that for every $t\in {\bar{I}}$, the set ${\mathcal {A}}_t:=\{u(t): u\in {\mathcal {A}}\}$ has compact closure in ${\mathbb {S}}$,

• (ii) $\underset{\delta \rightarrow 0}{\lim }\, \underset{u\in {\mathcal {A}}}{\sup }\, w_{[0,T]; {\mathbb {S}}}(u,\delta )=0$.

Let us recall the path space ${\mathcal {Z}}$:

$$\begin{aligned} {\mathcal {Z}}:= {\mathbb {D}}([0,T]; W^{-1,p^\prime })\cap {\mathbb {D}}([0,T]; L^2_w)\cap L^2_w(0,T; W^{1,2}) \cap L^2(0,T; L^2) \end{aligned}$$

equipped with the supremum topology ${\mathcal {T}}$; see (3.10). The following lemma provides a criterion for the relative compactness of a subset of $({\mathcal {Z}}, {\mathcal {T}})$.

Lemma 5.2

A set ${\mathcal {K}}\subset {\mathcal {Z}}$ is ${\mathcal {T}}$-relatively compact if the following three conditions hold:

• (a) for all $u\in {\mathcal {K}}$ and $t\in [0,T]$, $u(t)\in L^2$ and $\underset{u\in {\mathcal {K}}}{\sup }\, \underset{s\in [0,T]}{\sup }\, \Vert u(s)\Vert _{L^2} < + \infty$,

• (b) ${\mathcal {K}}$ is bounded in $L^2(0,T;W^{1,2})$ i.e., $\underset{u\in {\mathcal {K}}}{\sup }\, \int _0^T \Vert u(s)\Vert _{W^{1,2}}\,ds < + \infty$,

• (c) $\underset{\delta \rightarrow 0}{\lim }\, \underset{u\in {\mathcal {K}}}{\sup }\, w_{[0,T]; W^{-1,p^\prime }}(u,\delta )=0$.

Proof

Without loss of generality, we assume that ${\mathcal {K}}$ is closed in ${\mathcal {Z}}$. We first show that the compactness of ${\mathcal {K}}\subset {\mathcal {Z}}$ is equivalent to its sequential compactness. Indeed, thanks to the condition $\mathrm{b)}$, the set ${\mathcal {K}}$ is bounded in $L^2(0,T; W^{1,2})$ and therefore the weak topology ${\mathcal {T}}_2$ on $L^2_w(0,T; W^{1,2})$ induced on ${\mathcal {Z}}$ is metrizable. For any $r\in {\mathbb {R}}^*$, consider the ball ${\mathbb {B}}$

$$\begin{aligned} {\mathbb {B}}:=\{ y\in L^2: \Vert y\Vert _{L^2} \le r\}\,. \end{aligned}$$

Thanks to [4], the ball ${\mathbb {B}}$ endowed with the weak topology, denoted by ${\mathbb {B}}_w$, is metrizable. Consider the space ${\mathbb {D}}([0,T]; {\mathbb {B}}_w)$, the space of weakly càdlàg functions $y:[0,T]\rightarrow L^2$ with $\underset{t\in [0,T]}{\sup }\, \Vert y(t)\Vert _{L^2} \le r$. It is well-known that the space ${\mathbb {D}}([0,T]; {\mathbb {B}}_w)$ is metrizable by the metric $\delta _{T,r}$

$$\begin{aligned} \delta _{T,r}(y_1, y_2)&:= \inf _{\lambda \in \Lambda _T}\Bigg \{ \sup _{t\in [0,T]} \mathtt{q}_r \big (y_1(t), y_2(\lambda (t))\big ) + \sup _{t\in [0,T]} \big | t-\lambda (t)\big | \\&\quad + \sup _{s\ne t} \Big | \log \frac{\lambda (t)-\lambda (s)}{t-s}\Big | \Bigg \}\,, \end{aligned}$$

where $\mathtt{q}_r$ denote the metric compatible with the weak topology on the ball ${\mathbb {B}}$ and $\Lambda _T$ is the set of increasing homeomorphisms of [0, T]. Moreover, $\big ( {\mathbb {D}}([0,T]; {\mathbb {B}}_w), \delta _{T,r}\big )$ is a complete metric space. Therefore, In view of the assumption $\mathrm{a)}$, we may consider the metric subspace ${\mathbb {D}}([0,T]; {\mathbb {B}}_w)\subset {\mathbb {D}}([0,T]; L^2_w)$ with $r: =\underset{u\in {\mathcal {K}}}{\sup }\, \underset{s\in [0,T]}{\sup }\,\Vert u(s)\Vert _{L^2}$. Clearly, remaining spaces are all metrizable. Hence we conclude that the compactness of ${\mathcal {K}}\subset {\mathcal {Z}}$ is equivalent to its sequential compactness.

Let $\{u_n\}$ be a sequence in ${\mathcal {K}}$. Thanks to the Banach-Alaoglu theorem and the assumption $\mathrm{b)}$, the set ${\mathcal {K}}$ is compact in $L_w^2(0,T; W^{1,2})$. Next we show that $\{u_n\}$ is compact in ${\mathbb {D}}([0,T]; W^{-1,p^\prime })$. To show this, we need to check the conditions of Theorem 5.1 for ${\mathbb {S}}= W^{-1,p^\prime }$. By using condition $\mathrm{a)}$, we see that for every $t\in [0,T]$, the set $\{ u_n(t): n\in {\mathbb {N}}\}$ is bounded in $L^2$. Since the embedding $L^2 \subset W^{-1, p^\prime }$ is compact, the set $\{ u_n(t): n\in {\mathbb {N}}\}$ has compact closure in $W^{-1, p^\prime }$ for every $t\in [0,T]$. Recalling the condition $\mathrm{b)}$ together with the fact that the set $\{ u_n(t): n\in {\mathbb {N}}\}$ has compact closure in $W^{-1, p^\prime }$ for every $t\in [0,T]$, we conclude from Theorem 5.1 that the sequence $\{u_n\}$ is compact in ${\mathbb {D}}([0,T]; W^{-1,p^\prime })$. Therefore, there exist $u\in {\mathbb {D}}([0,T]; W^{-1,p^\prime })\cap L^2_w(0,T; W^{1,2})$ and a subsequence $\{u_{n_k}\}$ of $\{u_n\}$ such that

$$\begin{aligned} u_{n_k} \rightarrow u ~~~\text {in}~~{\mathbb {D}}([0,T]; W^{-1,p^\prime })\cap L^2_w(0,T; W^{1,2})~~~ \text {as}~~k\rightarrow \infty . \end{aligned}$$

Since $u_{n_k} \rightarrow u$ in ${\mathbb {D}}([0,T]; W^{-1,p^\prime })$, one has $u_{n_k}(t) \rightarrow u(t)$ in $W^{-1,p^\prime }$ for all continuity point t of u; see [1]. In view of the embedding $L^2\hookrightarrow W^{-1,p^\prime }$ and the condition $\mathrm{a)}$ together with the Lebesgue dominated convergence theorem, we get that

$$\begin{aligned} u_{n_k}\rightarrow u~~~\text {in} ~~L^2(0,T; W^{-1,p^\prime }) ~~~\text {as}~~k\rightarrow \infty . \end{aligned}$$

(4.6)

We now prove that $u_{n_k}\rightarrow u$ in $L^2(0,T; L^2)$. To do so, we proceed as follows. For each $\varepsilon >0$, there exists a constant $C(\varepsilon ) >0$ such that

$$\begin{aligned} \Vert y\Vert _{L^2}^2 \le \varepsilon \Vert y\Vert _{W^{1,2}}^2 + C(\varepsilon ) \Vert y\Vert _{W^{-1,p^\prime }}^2 \quad \forall ~y\in W^{1,2}\,. \end{aligned}$$

(4.7)

Replacing y by $u_{n_k}(s)-u(s)$ for almost all $s\in [0,T]$ in (4.7), and then integrating over the time interval [0, T], we get

$$\begin{aligned} \Vert u_{n_k}-u\Vert _{L^2(0,T; L^2)}^2&\le \varepsilon \Vert u_{n_k}-u\Vert _{L^2(0,T;W^{1,2})}^2 + C(\varepsilon ) \Vert u_{n_k}-u\Vert _{L^2(0,T; W^{-1,p^\prime })}^2 \nonumber \\&\le 2\varepsilon \big ( \Vert u_{n_k}\Vert _{L^2(0,T;W^{1,2})}^2 + \Vert u\Vert _{L^2(0,T;W^{1,2})}^2\big ) \nonumber \\&\quad + C(\varepsilon ) \Vert u_{n_k}-u\Vert _{L^2(0,T; W^{-1,p^\prime })}^2 \nonumber \\&\le 2\varepsilon {\tilde{C}} + C(\varepsilon ) \Vert u_{n_k}-u\Vert _{L^2(0,T; W^{-1,p^\prime })}^2\, \end{aligned}$$

(4.8)

where ${\tilde{C}}= \underset{y \in {\mathcal {K}}}{\sup }\, \int _0^T \Vert y(t)\Vert _{W^{1,2}}^2\,dt$, which is finite by the condition $\mathrm{b)}$. Passing to the limit in (4.8) as $k\rightarrow \infty$, we have, thanks to (4.6) and the arbitrariness of $\varepsilon$,

$$\begin{aligned} \Vert u_{n_k}-u\Vert _{L^2(0,T; L^2)}^2 \rightarrow 0 ~~~\text {as}~~k\rightarrow \infty . \end{aligned}$$

It remains to show that the sequence $\{u_n\}$ is compact in ${\mathbb {D}}([0,T];{\mathbb {B}}_w)$. Since $\{u_n\}$ is compact in ${\mathbb {D}}([0,T]; W^{-1,p^\prime })$, and by the condition $\mathrm{a)}$, $u_n$ satisfies the estimate

$$\begin{aligned} \sup _{n} \sup _{s\in [0,T]} \Vert u_n(s)\Vert _{L^2} \le r: =\underset{u\in {\mathcal {K}}}{\sup }\, \underset{s\in [0,T]}{\sup }\,\Vert u(s)\Vert _{L^2}, \end{aligned}$$

we get from Lemma 2 in [25] that $u_n, u \in {\mathbb {D}}([0,T];{\mathbb {B}}_w)$ and $u_n \rightarrow 0$ in ${\mathbb {D}}([0,T];{\mathbb {B}}_w)$ as $n\rightarrow \infty$. This completes the proof. $\square$

1.1 Proof of Theorem 3.5

Let ${\mathcal {L}}(X_\tau )$ be the laws of the sequence $(X_\tau )$ on ${\mathcal {Z}}$. We need to show that for every $\varepsilon >0$, there exists a compact set $K_\varepsilon$ of ${\mathcal {Z}}$ such that

$$\begin{aligned} \sup _{\varepsilon >0} {\mathcal {L}}(X_\tau )(K_\varepsilon ) \ge 1-\varepsilon . \end{aligned}$$

Thanks to the Markov inequality and the assumption $\mathrm{(i)}$,

$$\begin{aligned} {\mathbb {P}} \Big ( \sup _{t\in [0,T]} \Vert X_\tau (t)\Vert _{L^2}> R_1\Big )&\le \frac{{\mathbb {E}}\Big [ \sup _{t\in [0,T]} \Vert X_\tau (t)\Vert _{L^2} \Big ]}{R_1} \\&\le \frac{ \sup _{\tau >0} {\mathbb {E}}\Big [ \sup _{t\in [0,T]}\Vert X_{\tau }(t)\Vert _{L^2}\Big ]}{R_1}\le \frac{C_1}{R_1} \,. \end{aligned}$$

By choosing $R_1>0$ such that $R_1> \frac{3C_1}{\varepsilon }$, we get

$$\begin{aligned} {\mathbb {P}} \Big ( \sup _{t\in [0,T]} \Vert X_\tau (t)\Vert _{L^2} > R_1\Big ) \le \frac{\varepsilon }{3}\,. \end{aligned}$$

(4.9)

Similarly, by using Markov-inequality, the assumption $\mathrm{(ii)}$ and the fact that $\Vert u\Vert _{W^{1,2}}\le C \Vert u\Vert _{W_0^{1,p}}$ for any $u\in W_0^{1,p}$, we have, for any $R_2>0$

$$\begin{aligned} {\mathbb {P}}\Big ( \Vert X_\tau \Vert _{L^2(0,T; W^{1,2})}> R_2\Big )&\le {\mathbb {P}}\Big ( \Vert X_\tau \Vert _{L^2(0,T; W_0^{1,p})}> \frac{R_2}{C}\Big ) \\&\le \frac{ C^2}{R_2^2} \, \sup _{\tau >0}{\mathbb {E}}\Big [ \Vert X_\tau \Vert _{L^2(0,T;W_0^{1,p})}^2\Big ] \le \frac{C_2 C^2}{R_2^2}\,. \end{aligned}$$

Let $R_2$ be such that $\frac{C_2 C^2}{R_2^2}\le \frac{\varepsilon }{3}$. Then

$$\begin{aligned} {\mathbb {P}}\Big ( \Vert X_\tau \Vert _{L^2(0,T; W^{1,2})} > R_2\Big ) \le \frac{\varepsilon }{3}\,. \end{aligned}$$

(4.10)

Since $(X_\tau )_{\tau >0}$ satisfies the Aldous condition in $W^{-1, p^\prime }$ (cf. condition $\mathrm{(iii)}$), $(X_\tau )$ also satisfies the following condition; see [16, Theorem 2.2.2]: for every $\varepsilon >0$ and $\gamma >0$, there exists a $\delta >0$ such that

$$\begin{aligned} \sup _{\tau>0} {\mathbb {P}}\Big ( w_{[0,T]; W^{-1, p^\prime }}(X_\tau , \delta ) > \gamma \Big ) \le \varepsilon . \end{aligned}$$

Then, thanks to [25, Lemma 7 ], for every $\varepsilon >0$ there exists a subset ${\mathcal {A}}_{\frac{\varepsilon }{3}} \subset {\mathbb {D}}([0,T]; W^{-1,p^\prime })$ such that

• (I) $\underset{\tau>0}{\sup }\, {\mathcal {L}}(X_\tau )({\mathcal {A}}_{\frac{\varepsilon }{3}}) > 1-\frac{ \varepsilon }{3}$,

• (II) $\underset{\delta \rightarrow 0}{\lim }\, \underset{u\in {\mathcal {A}}_{\frac{\varepsilon }{3}}}{\sup }\, w_{[0,T]; W^{-1, p^\prime }}(u, \delta )=0$.

Define the sets

$$\begin{aligned} {\mathcal {A}}_1:= \big \{ u \in {\mathcal {Z}}: \sup _{t\in [0,T]} \Vert u(t)\Vert _{L^2} \le R_1 \big \}; \quad {\mathcal {A}}_2:= \big \{ u \in {\mathcal {Z}}: \Vert u\Vert _{L^2(0,T; W^{1,2})} \le R_2 \big \}\,. \end{aligned}$$

Let $K_\varepsilon$ be the closure of the set ${\mathcal {A}}_1 \cap {\mathcal {A}}_2 \cap {\mathcal {A}}_{\frac{\varepsilon }{3}}$ in ${\mathcal {Z}}$. Then by Lemma 5.2, $K_\varepsilon$ is compact in ${\mathcal {Z}}$. Moreover, by (4.9), (4.10) and $\mathrm{(II)}$, one can easily deduce that

$$\begin{aligned} \sup _{\tau >0} {\mathcal {L}}(X_\tau )(K_\varepsilon ) \ge 1-\varepsilon . \end{aligned}$$

This completes the proof of Theorem 3.5 .