Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

Abstract

We study optimal control problems for stochastic nonlinear Schrödinger equations in both the mass subcritical and critical case. For general initial data of the minimal $L^2$ regularity, we prove the existence and first order Lagrange condition of an open loop control. In particular, these results apply to the stochastic nonlinear Schrödinger equations with the critical quintic and cubic nonlinearities in dimensions one and two, respectively. Furthermore, we obtain uniform estimates of (backward) stochastic solutions in new spaces of type $U^2$ and $V^2$, adapted to evolution operators related to linear Schrödinger equations with lower order perturbations. These estimates yield a new temporal regularity of (backward) stochastic solutions, which is crucial for the tightness of approximating controls induced by Ekeland’s variational principle.

Appendix

Proof of Lemma 3.8

Let $I_j=[jh,(j+1)h]$, $0\le j\le [\frac{T}{h}]-1=:L^*$. Since $v\in C([0,T]; H)$, there exists $t_j\in I_j$ such that $\sup _{t\in I_j} \Vert v(t+h)-v(t)\Vert _H = \Vert v(t_j+h)-v(t_j)\Vert _H$, $0\le j\le L^*$. Then,

$$\begin{aligned} \int _0^{T-h} \Vert v(t+h)-v(t)\Vert _H^p dt \le&\, h \sum \limits _{j=0}^{L^*} \Vert v(t_j+h)-v(t_j)\Vert _H^p. \end{aligned}$$

Moreover, letting $\{t_j'\}_{j=0}^{L} = \{t_j+h, t_j\}_{j=0}^{L^*}$ we have that

$$\begin{aligned} \sum \limits _{j=0}^{L^*} \Vert v(t_j+h)-v(t_j)\Vert _H^p \le 2^{p+1} \sum \limits _{j=0}^{L-1} \Vert v(t'_{j+1})-v(t'_j)\Vert _H^p \le 2^{p+1} \Vert v\Vert _{V^p(0,T)}^p. \end{aligned}$$

Thus, combining the estimates above we prove (3.9). $\square$

Proof of Proposition 3.9

First we prove (3.10). It suffices to prove the case that $V(t_0, \cdot ) u$ is a $U^q$-atom of the form $V(t_0, t) u(t) = \sum _{j=1}^n u_j {{{\mathcal {X}}}}_{[t_j, t_{j+1})}(t)$, $t\in I$, such that $\{t_j\}_{j=1}^n \subseteq I$, $\sum _{j=1}^n |u_j|_{L^2}^q = 1$. This yields that $u(t) = \sum _{j=1}^n V(t, t_0) u_j {{{\mathcal {X}}}}_{[t_j, t_{j+1})}(t)$, and so

$$\begin{aligned} \Vert u\Vert ^q_{L^q(I; L^p)} = \sum \limits _{j=1}^n \Vert V(\cdot , t_0)u_j\Vert ^q_{L^q(t_j, t_{j+1}; L^p)}. \end{aligned}$$

(7.1)

Using Theorem 3.1 and $\Vert V(t_j, t_0)\Vert _{{\mathcal {L}}(L^2, L^2)} \le C(T) \in L^\rho (\Omega )$ we have

$$\begin{aligned} \Vert V(\cdot , t_0)u_j\Vert ^q_{L^q(t_j, t_{j+1}; L^p)} =&\, \Vert V(\cdot , t_j) V(t_j, t_0)u_j\Vert ^q_{L^q(t_j, t_{j+1}; L^p)} \nonumber \\ \le&\, C_T^q |V(t_j, t_0) u_j |^q_{L^2} \le (C'(T))^q |u_j|^q_{L^2}, \end{aligned}$$

(7.2)

where $C'(T)$ is independent of (p, q) and $C'(T) \in L^\rho (\Omega )$ for any $1\le \rho <{\infty }$. Plugging (7.2) into (7.1) yields

$$\begin{aligned} \Vert u\Vert _{L^q(I; L^p)} \le C'(T) \left( \sum \limits _{j=1}^n|u_j|^q_{L^2}\right) ^{\frac{1}{q}} \le C'(T), \end{aligned}$$

(7.3)

which implies (3.10).

In order to prove (3.11), we see that for any partition $\{t_j\}_{j=0}^m \in {\mathcal {Z}}$, since $\Vert V(t_0,t_{j-1})\Vert _{{\mathcal {L}}(L^2, L^2)} \le C(T) \in L^\rho (\Omega )$, if $\widetilde{f}:= {{\mathcal {X}}}_I f$,

$$\begin{aligned}&\bigg |\int _{t_{j-1}}^{t_{j}} V(t_0, s) \widetilde{f}(s) ds \bigg |_{L^2}\nonumber \\&\quad \le C(T) \bigg |\int _{t_{j-1}}^{t_{j}} V(t_{j-1}, s) \widetilde{f}(s) ds \bigg |_{L^2} \nonumber \\&\quad = C(T) \sup \limits _{|z|_{L^2}\le 1} \left<\int _{t_{j-1}}^{t_{j}} V(t_{j-1}, s) \widetilde{f}(s) ds, z\right>_2 \nonumber \\&\quad \le C(T) \Vert \widetilde{f}\Vert _{L^{q'}(t_{j-1}, t_{j}; L^{p'})} \sup \limits _{|z|_{L^2}\le 1} \Vert V^*(t_{j-1}, \cdot ) {{\mathcal {X}}}_I z\Vert _{L^q(t_{j-1}, t_{j}; L^p)}. \end{aligned}$$

(7.4)

Estimating as in [63, Lemma 5.3], we have

$$\begin{aligned} \Vert V^*(t_{j-1}, \cdot ) {{\mathcal {X}}}_I z\Vert _{L^q(t_{j-1}, t_{j}; L^p) \cap L^2(t_{j-1}, t_{j}; H^\frac{1}{2}_{-1})} \le C'(T) |z|_{L^2} \le C'(T), \end{aligned}$$

(7.5)

where $C'(T) \in L^\rho (\Omega )$ for any $1\le \rho <{\infty }$. Then, we get

$$\begin{aligned} \sum \limits _{j=1}^m \bigg |\int _{t_{j-1}}^{t_{j}} V(t_0, s) \widetilde{f}(s) ds \bigg |^{q'}_{L^2} \le&\, (C''(T))^{q'} \sum \limits _{j=1}^m \Vert \widetilde{f}\Vert ^{q'}_{L^{q'}(t_{j-1}, t_{j}; L^{p'})} \nonumber \\ =&\, (C''(T))^{q'} \Vert f\Vert ^{q'}_{L^{q'}(I; L^{p'})}, \end{aligned}$$

(7.6)

where $C''(T)\in L^\rho (\Omega )$, $\forall 1\le \rho <{\infty }$, independent of (p, q). This implies (3.11).

Regarding (3.12), let $f = f_1+f_2$ with $f_1\in L^{q'}(I;L^{p'})$, $f_2 \in L^2(I; H^{-\frac{1}{2}}_1)$, and set $\widetilde{f}_j = {{\mathcal {X}}}_I f_j$, $j=1,2$. We can take a finer partition $\{t_j\}_{j=0}^m$ such that $\Vert f\Vert _{L^{q'}(t_{j-1}, t_{j}; L^{p'})} \le 1$, $1\le j\le m$. Then, estimating as in (7.6), since $q>2$, $q'<2$, we have

$$\begin{aligned}&\sum \limits _{j=1}^m \bigg |\int _{t_{j-1}}^{t_{j}} V(t_0, s) \widetilde{f}_1(s) ds \bigg |^{2}_{L^2}\\&\quad \le (C''(T))^{2} \sum \limits _{j=1}^m \Vert \widetilde{f}_1\Vert ^{2}_{L^{q'}(t_{j-1}, t_{j}; L^{p'})} \le (C''(T))^{2} \Vert f_1\Vert ^{q'}_{L^{q'}(I; L^{p'})}, \end{aligned}$$

which implies that

$$\begin{aligned} \bigg \Vert \int _{t_0}^\cdot V(t_0, s) f_1(s) ds \bigg \Vert _{V^2(I)} \le C''(T) \Vert f_1\Vert ^{\frac{q'}{2}}_{L^{q'}(I; L^{p'})} \le C''(T) (1+ \Vert f_1\Vert _{L^{q'}(I; L^{p'})}). \end{aligned}$$

(7.7)

Moreover, arguing as in the proof of (7.4) and using (7.5) we have

$$\begin{aligned} \sum \limits _{j=1}^m \bigg |\int _{t_{j-1}}^{t_{j}} V(t_0, s) \widetilde{f}_2(s) ds \bigg |^2_{L^2} \le (C''(T))^2 \Vert f_2\Vert ^2_{L^{2}(I; H^{-\frac{1}{2}}_1)}. \end{aligned}$$

This yields that

$$\begin{aligned} \bigg \Vert \int _{t_0}^\cdot V(t_0, s) f_2(s) ds \bigg \Vert _{V^2(I)} \le C''(T) \Vert f_2\Vert _{L^{2}(I; H^{-\frac{1}{2}}_1)}. \end{aligned}$$

(7.8)

Therefore, combining (7.7) and (7.8) together we prove (3.12). $\square$

In order to prove Theorem 3.13, we first prove the short-time perturbation result below as in [64].

Proposition 7.1

(Mass-Critical Short-time Perturbation). Consider the situations in Theorem 3.13. Assume also the smallness conditions

$$\begin{aligned}&|I|+ \Vert \widetilde{v}\Vert _{L^{q}(I; L^p)} \le \delta , \end{aligned}$$

(7.9)

$$\begin{aligned}&\Vert V(\cdot , t_0)(v(t_0)-\widetilde{v}(t_0) -R(t_0)) \Vert _{L^{q}(I; L^p)} \le {\varepsilon }, \ \Vert R\Vert _{L^{q}(I; L^p) \cap L^1(I; L^2)} \le {\varepsilon },\ \end{aligned}$$

(7.10)

$$\begin{aligned}&\Vert e \Vert _{{N}^0(I) + L^2(I; H^{- \frac{1}{2}}_{1})} \le {\varepsilon } \end{aligned}$$

(7.11)

for some $0<{\varepsilon }\le \delta$ where $\delta = \delta (C_T)>0$ is a small constant, and $C_T$ is as in Theorem 3.13. Then, we have

$$\begin{aligned}&\Vert v-\widetilde{v} -R\Vert _{L^{q}(I; L^p) \cap C(I; L^2)} \le C(C_T) {\varepsilon }, \end{aligned}$$

(7.12)

$$\begin{aligned}&\Vert i (F(v) - F(\widetilde{v})) + G(v- \widetilde{v} - R) \Vert _{{N}^0(I)} \le C(C_T) {\varepsilon }, \end{aligned}$$

(7.13)

where $(\delta (C_T))^{-1}$, $C(C_T)$ can be taken to nondecreasing with respect to $C_T$.

Proof

We mainly prove Proposition 7.1 for the case where p satisfies that $\frac{1}{p} \in (\max \{\frac{1}{2{\alpha }}, \frac{1}{2} - \frac{1}{2d}\}, \frac{1}{{\alpha }}(\frac{1}{2} + \frac{1}{d}))$ with $1\le d\le 3$, ${\alpha }=1+\frac{4}{d}$. The case $p =2+\frac{4}{d}$ with $d\ge 1$ can be proved similarly.

As mentioned below Hypothesis $(H0)^*$, there exists another Strichartz pair $(\widetilde{p}, \widetilde{q})$ such that $(\frac{1}{\widetilde{p}'}, \frac{1}{\widetilde{q}'}) = (\frac{{\alpha }}{p}, \frac{{\alpha }}{q})$, where $q\in (2,{\infty })$ is such that (p, q) is a Strichartz pair, and $\widetilde{p}', \widetilde{q}'$ are the conjugate numbers of $\widetilde{p}, \widetilde{q}$ respectively.

Let $z:= v -\widetilde{v} - R$, $F(\widetilde{v}):=|\widetilde{v}|^{\frac{4}{d}} \widetilde{v}$ and $F(z +R+ \widetilde{v})$ be defined similarly. By equations (3.17) and (3.18),

$$\begin{aligned} z(t) =&\, V(t,t_0)z(t_0) + \int _{t_0}^t V(t,s) \big (i (F(z + R+\widetilde{v}) - F(\widetilde{v})) + Gz + GR + e \big ) ds. \end{aligned}$$

(7.14)

We set $S(I):= \Vert i (F(z + R+\widetilde{v}) - F(\widetilde{v})) +Gz\Vert _{ N^0(I)}$. By Hölder’s inequality and (7.9)–(7.11),

$$\begin{aligned} S(I) \le&\, \Vert (F(z+R+\widetilde{v}) - F(\widetilde{v}))\Vert _{L^{\widetilde{q}'}(0,T; L^{\widetilde{p}'})} + \Vert Gz\Vert _{L^1(0,T;L^2)} \nonumber \\ \le&\, C\left( \Vert \widetilde{v}\Vert _{L^q(I; L^p)}^{\frac{4}{d}} \Vert z + R\Vert _{L^q(I; L^p)} + \Vert z + R\Vert _{L^q(I; L^p)}^{1+\frac{4}{d}}\right) + \Vert Gz\Vert _{L^1(I; L^2)} \nonumber \\ \le&\, C_1\left( \delta ^{\frac{4}{d}}{\varepsilon }+ \delta ^{\frac{4}{d}} \Vert z\Vert _{L^q(I; L^p)} + \delta \Vert z\Vert _{C(I; L^2)} + \Vert z\Vert _{L^q(I; L^p)}^{1+\frac{4}{d}}\right) , \end{aligned}$$

(7.15)

where $C_1(\ge 1)$ depends on d and $\Vert G\Vert _{L^{\infty }}(I\times {{\mathbb {R}}}^d)$. Moreover, applying Theorem 3.1 to (7.14) and using (7.10) and (7.11) we have

$$\begin{aligned} \Vert z\Vert _{L^q(I; L^p) \cap C(I; L^2)} \le&\, C_T (\Vert V(\cdot , t_0)z(t_0)\Vert _{L^q(I; L^p)} + S(I) \nonumber \\&\quad + \Vert GR\Vert _{L^1(I; L^2)} + \Vert e\Vert _{N^0(I) + L^2(I; H^{- \frac{1}{2}}_{1})}) \nonumber \\ \le&\, C_2C_T ({\varepsilon }+ S(I)), \end{aligned}$$

(7.16)

where $C_2$ depends on $\Vert G\Vert _{L^{\infty }}(I\times {{\mathbb {R}}}^d)$. Then, combining (7.15), (7.16) we get

$$\begin{aligned} \Vert z\Vert _{L^{q}(I; L^p)\cap C(I; L^2)} \le&\, C_1C_2C_T \big (2{\varepsilon }+ (\delta ^{\frac{4}{d}} + \delta ) \Vert z\Vert _{L^q(I; L^p)\cap C(I; L^2)} + \Vert z\Vert _{L^q(I; L^p)}^{1+\frac{4}{d}}\big ). \end{aligned}$$

Thus, in view of Lemma 6.1 in [8], taking $\delta = \delta (C_T)$ small enough such that $C_1C_2C_T(\delta ^{\frac{4}{d}} + \delta ) \le \frac{1}{2}$ and $4C_1C_2C_T \delta < (1-\frac{1}{{\alpha }})(2{\alpha }C_1C_2C_T)^{-\frac{1}{{\alpha }-1}}$ we obtain (7.12). Moreover, plugging (7.12) into (7.15) we obtain (7.13).

Therefore, the proof is complete. $\square$

Proof of Theorem 3.13

First fix $\delta = \delta (C_T)$ as in Proposition 7.1. We divide $[t_0,T]$ into finitely many subintervals $[t'_j,t'_{j+1}]$, $0\le j\le l'$, such that $t'_{j+1} = \inf \{t>t'_j; \Vert \widetilde{v}\Vert _{L^{q}(t'_j,t; L^p)} =\frac{\delta }{2}\} \wedge T$. Then, $l'\le (2L/\delta )^{2+\frac{4}{d}}<{\infty }$.

Take a new partition $\{I_j\}_{j=0}^l:= \{[t_j, t_{j+1}]\}_{j=0}^l$ of [0, T], such that $\{t_j; 0\le j\le l+1\} = \{t'_j; 0\le j\le l'+1\} \cup \{t_0 + \frac{j\delta }{2}; 0\le j\le [\frac{2(T-t_0)}{\delta }]\}$. Then,

$$\begin{aligned} l\le (2L/\delta )^{2+\frac{4}{d}} + [2(T-t_0)/\delta ], \ \ |I_j| + \Vert \widetilde{v}\Vert _{L^{q}(I_j; L^p)} \le \delta ,\ \ 0\le j\le l. \end{aligned}$$

Let $C(0) = C(C_T)$, $C(j+1) = C(0)C_T^2 (\sum _{k=0}^j C(k)+ \Vert G\Vert _{L^{\infty }}$$+2)$, $0\le j\le l-1$, where $C(C_T)$ and $C_T$ are the constants in Proposition 7.1 and Theorem 3.1, respectively. Choose ${\varepsilon }_*= {\varepsilon }_*(C_T, L)$ sufficiently small such that

$$\begin{aligned} C_T^2 \left( \sum \limits _{k=0}^l C(k) +\Vert G\Vert _{L^{\infty }} + 2\right) {\varepsilon }_* \le \delta . \end{aligned}$$

(7.17)

We claim that (7.12) and (7.13) hold on $I_j$ with C(j) replacing $C(C_T)$ for every $0\le j\le l$.

To this end, we first see that Proposition 7.1 yields the claim for $j=0$. Suppose that the claim is also valid for each $0\le k\le j<l$. We shall use Proposition 7.1 to show that it also holds on $I_{j+1}$.

For this purpose, applying Theorem 3.1 to (7.14) again and using (3.20), (7.17) and the inductive assumption we have

$$\begin{aligned}&\Vert V(\cdot , t_{j+1})(v(t_{j+1}) - \widetilde{v}(t_{j+1}) -R(t_{j+1}))\Vert _{L^{q}(I_{j+1}; L^p)} \\&\quad \le \Vert V(\cdot , t_0)(v(t_0)-\widetilde{v}(t_0))\Vert _{L^{q}(I; L^p)} \\&\quad + C^2_T \big ( S(t_0, t_{j+1}) + \Vert GR\Vert _{L^1(t_0,t_{j+1};L^2)} + \Vert e\Vert _{N^0(t_0, t_{j+1}) + L^2(t_0,t_{j+1}; H^{-\frac{1}{2}}_1)} \big ) \\&\quad \le {\varepsilon }+ C_T^2 \big (\sum \limits _{k=0}^j C(k) {\varepsilon }+ \Vert G\Vert _{L^{\infty }(I\times {{\mathbb {R}}}^d)}{\varepsilon }+{\varepsilon }\big ) \le \delta . \end{aligned}$$

Thus, the conditions (7.9)–(7.11) of Proposition 7.1 are satisfied with $C_T^2$$(\sum _{k=0}^j C(k)+ \Vert G\Vert _{L^{\infty }}+2){\varepsilon }$ replacing ${\varepsilon }$. Applying Proposition 7.1 we prove the claim on $I_{j+1}$.

Therefore, using inductive arguments we prove the claim on $I_j$ for every $0\le j\le l$, thereby proving Theorem 3.13. The proof is complete. $\square$

Proof of (4.14)

The proof is similar to that in [32]. Without loss of generality, we may assume $\rho \ge q$. By Minkowski’s inequality and Burkholder’s inequality,

$$\begin{aligned} \Vert M_1^*\Vert _{L^\rho (\Omega ; L^1(0,T))} \le&\, C \Vert M_1^*\Vert _{L^1(0,T; L^\rho (\Omega ))} \nonumber \\ \le&\, C \bigg \Vert \bigg |\int _0^\cdot \Vert e^{-i(\cdot -s)\Delta } X(s)\Phi \Vert ^2_{HS({{\mathbb {R}}}^N; L^2)} ds\bigg |^{\frac{1}{2}} \bigg \Vert _{L^1(0,T)}, \end{aligned}$$

(7.18)

where the operator $\Phi : {{\mathbb {R}}}^N \mapsto L^{\infty }$ is defined by $\Phi (x) = \sum _{j=1}^N \mu _j e_j x_j$ for $x= (x_1,\ldots , x_N)$, and $\Vert \cdot \Vert _{HS({{\mathbb {R}}}^N; L^2)}$ denotes the Hilbert-Schmidt norm (see, e.g., [51]). Note that,

$$\begin{aligned} \Vert e^{-i(t -s)\Delta } X(s)\Phi \Vert ^2_{HS({{\mathbb {R}}}^N; L^2)} \le&\, C \sum \limits _{j=1}^N |\mu _j|^2|e_j|^2_{L^{\infty }} |X_0|^2_{L^2}. \end{aligned}$$

Plugging this into (7.18) and taking into account that the upper bound is independent of $u\in {{\mathcal {U}}}_{ad}$ we obtain $\sup _{u\in {{\mathcal {U}}}_{ad}}\Vert M_1^*\Vert _{L^\rho (\Omega ; L^1(0,T))} <{\infty }$.

Similarly, again by Minkowski’s inequality and Burkholder’s inequality,

$$\begin{aligned} \Vert M_2^*\Vert _{L^\rho (\Omega ; L^q(0,T))} \le&\, \Vert M_2^*\Vert _{L^q(0,T; L^\rho (\Omega ))} \nonumber \\ \le&\, C(\rho , T) \bigg \Vert \bigg |\int _0^\cdot \Vert e^{-i(\cdot -s)\Delta } X(s)\Phi \Vert ^2_{{\mathcal {R}}({{\mathbb {R}}}^N; L^p)} ds\bigg |^{\frac{1}{2}} \bigg \Vert _{L^q(0,T)}, \end{aligned}$$

(7.19)

where $\Vert \cdot \Vert _{{\mathcal {R}}({{\mathbb {R}}}^N; L^p)}$ denotes the ${\gamma }$-radonifying norm (see, e.g., [20, 22, 32]).

Note that, by the dispersive inequality $\Vert e^{-i(t-s)\Delta }\Vert _{{\mathcal {L}}(L^{p'}, L^p)} \le C (t-s)^{-\frac{d}{2}(1-\frac{2}{p})}$,

$$\begin{aligned}&\Vert e^{-i(t -s)\Delta } X(s)\Phi \Vert _{{\mathcal {R}}({{\mathbb {R}}}^N; L^p)}\\&\quad \le \Vert \Phi \Vert _{{\mathcal {R}}({{\mathbb {R}}}^N; L^{\frac{2p}{p-2}})} \Vert X(s)\Vert _{{\mathcal {L}}(L^{\frac{2p}{p-2}}, L^{p'})} \Vert e^{-i(t-s)\Delta }\Vert _{{\mathcal {L}}(L^{p'}; L^p)} \\&\quad \le C(N)|X_0|_{L^2} \left( \sum \limits _{j=1}^N |\mu _j|^2 |e_j|^2_{L^{\frac{2p}{p-2}}}\right) ^\frac{1}{2} (t-s)^{-\frac{d}{2}(1-\frac{2}{p})}. \end{aligned}$$

Therefore, since the upper bound is independent of $u\in {{\mathcal {U}}}_{ad}$, using (7.19) we obtain that $\sup _{u\in {{\mathcal {U}}}_{ad}} \Vert M_2^*\Vert _{L^\rho (\Omega ; L^q(0,T))} <{\infty }$ if $p<\frac{2d}{d-1}$ and so finish the proof. $\square$

Theorem 7.2

([31, Theorem 1], see also [30]) Let V be a complete metric space, and $F:V \mapsto {{\mathbb {R}}}\cup \{+{\infty }\}$ a l.s.c. function, $\not \equiv +{\infty }$, bounded from below. Let ${\varepsilon }>0$ be given, and a point $u\in V$ such that $F(u) \le \inf _{V}F +{\varepsilon }$. Then, there exists some point $v\in V$ such that $F(v) \le F(u)$, $d(u,v) \le 1$, and

$$\begin{aligned} F(w)> F(v) - {\varepsilon }d(v,w),\ \ \forall w\not =v. \end{aligned}$$