## Banach空间中微分变分不等式系统的Bang-Bang准则

1 桂林理工大学南宁分校, 南宁 530001

2 玉林师范学院广西高校复杂系统优化与大数据处理重点实验室, 广西 玉林 537000

3 玉林师范学院数学与统计学院, 广西 玉林 537000

## On the Bang-Bang Principle for Differential Variational Inequalities in Banach Spaces

Shi Cuiyun1, Bin Maojun,2,3

1 Guilin University of Technology at Nanning, Nanning 530001

2 Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Guangxi Yulin 537000

3 School of Mathematics and Statistics, Yulin Normal University, Guangxi Yulin 537000

 基金资助: 广西自然科学基金.  2020GXNSFAA159152广西自然科学基金.  2020GXNSFBA297142广西自然科学基金.  2021GXNSFAA220130广西自然科学基金.  2022GXNSFAA035617

 Fund supported: the NSF of Guangxi.  2020GXNSFAA159152the NSF of Guangxi.  2020GXNSFBA297142the NSF of Guangxi.  2021GXNSFAA220130the NSF of Guangxi.  2022GXNSFAA035617

Abstract

In this paper, we discuss a class of differential variational inequalities systems, which are obtained by semilinear evolution equations and generalized variational inequalities. At first, we consider the properties of solution set for generalized variational inequalities. Secondly, the existence results are shown by fixed point method for semilinear differential variational inequality. Our approaches are based on semigroup theory and fixed point theorem. Moreover, by using the density results, the nonlinear and infinite dimensional versions of the "bang-bang" principle for differential variational inequalities systems is derived. Also, an obstacle parabolic-elliptic system is given to illustrate the application of the obtained theory.

Keywords： Differential variational inequalities ; "Bang-Bang" principle ; Density ; KKM mapping

Shi Cuiyun, Bin Maojun. On the Bang-Bang Principle for Differential Variational Inequalities in Banach Spaces. Acta Mathematica Scientia[J], 2022, 42(6): 1653-1670 doi:

## 1 引言

(ii) $F$在点$z_0$处是$h$ -下半连续的, 若$z\mapsto h^*(z_0, z)$在点$z_0$处连续; 若$F$在每一点$z_0\in V$处都是$h$ - 下半连续的, 则称$F $$h - 下半连续的; (iii) F 在点 z_0 处是 h -连续的, 若 F 在点 z_0 处既是 h -上半连续的又是 h -下半连续的; 若 F 在每一点 z_0\in V 处都是 h - 连续的, 则称 F$$ h$ - 连续的.

$F:J\rightarrow {\cal P}(X)$为集值映射. 如果$F^{-1}(E)=\{t\in J:F(t)\cap E\neq\emptyset\}\in \Sigma$对于每个闭子集$E\subseteq X$, 集值映射$F:J\rightarrow {\cal P}_f(X)$是可测的. 若$F:T\times X\rightarrow {\cal P}_f(X)$, 则集值映射$F$可测定义为$F^{-1}(E)\in\Sigma\otimes{\cal B}_{X}$, 其中$\Sigma\otimes{\cal B}_{X}$表示由积集族$A\times B$生成的$\sigma$ -代数, $A\in\Sigma$, $B\in{\cal B}_{X}$, 及${\cal B}_{X} $$X 中Borel集生成的 \sigma -代数[16]. 引理2.1[47,引理 3.2] 设 G:J\rightarrow {\cal P}(X) 是可测集值映射且 u:J\rightarrow X 是可测函数, 则对任意的可测函数 r:J\rightarrow (0, +\infty) , 存在 G 中的可测选择函数 g 使得a.e. t\in J , X, Z 是两个Hausdorff拓扑空间, F: X\rightarrow {\cal P}(Z) . F$$ x_0\in X$点称之为下半连续的(简记为l.s.c.), 如果对于每个开子集$D\subseteq Z$, $F(x_0)\cap D\neq\emptyset$, 存在$x_0$的邻域$O(x_0)$使得$F(x)\cap D\neq\emptyset$, 对所有的$x\in O(x_0)$. $F $$x_0\in X 点称之为上半连续的(简记为u.s.c.), 如果对于每个开子集 D\subseteq Z , F(x_0)\subseteq D , 存在 x_0 的邻域 O(x_0) 使得 F(x)\subseteq D , 对于所有的 x\in O(x_0) . 对于l.s.c和u.s.c的性质, 参见文献[17]. 引理2.2[18, 命题 1.3.1] 假设 E$$ V$是两个Banach空间, 如果$F:J\times E\rightarrow {\cal P}_{k}( V)$满足卡氏条件(Carathéodory condition) 或者$F$是半连续的(上半连续或下半连续), 那么集值函数$F$是叠合可测的.

$$$\|u(\cdot)\|_{\omega}=\sup\limits_{0\leq t_1\leq t_2\leq b}\Big\|\int_{t_1}^{t_2}u(s){\rm d}s\Big\|_X, \, \, \mbox{对于 }\ u\in L^q(J, X).$$$

$\begin{eqnarray} x(t)=T(t)x_{0} +\int_{0}^{t}T(t-s)B(s, x(x))u(s){\rm d}s, \; t\in J. \end{eqnarray}$

(i) $G$是KKM映射, 即: 对$\forall \{v_1, v_2, \cdots, v_n\}\subset K$, 其凸包co$\{v_1, v_2, \cdots, v_n\}\subseteq\bigcup\limits^n_{i=1}G(v_i)$;

(ii) 对于每一个$v\in K$, $G(v) $$V 中的闭集; (iii) 对于某一个 v_0\in K , G(v_0)$$ V$是紧的.

($\varphi_3$) 对所有的$v\in K, \varphi(\cdot, v)$是弱u.s.c.;

($\varphi_4$) 对所有的$x\in K, \varphi(x, \cdot)$是凸的.

(i) $G: Y\rightarrow Y^* $$K 中是单调和 h -半连续的; (ii) \varphi:Y\rightarrow \overline{{{\Bbb R}} } 满足假设 (\varphi_1)-(\varphi_4) . u\in {\rm SOL}(K, w+G(\cdot), \varphi) 当且仅当 $$\langle w+G(v), v-u\rangle+\varphi(v, u) \geq0, \quad\forall v\in K.$$ 另外, 如果 K$$ Y$中是无界的, 则存在$u_0\in K $$d>0 使得 对所有的 v\in K$$ \|v\|_Y>d$, 则(2.4)式的解在$Y$中是非空闭凸的.

$$$\limsup\limits_{u\in K, \; \|u\|_{Y}\rightarrow \infty}\frac{\langle G(u), u-v^*\rangle+\varphi(v^*, u)}{\|u\|_{Y}}\rightarrow +\infty,$$$

用反正法证明. 假设存在序列$\{u_n\}\subset$ SOL$(K, w+G(\cdot), \varphi)$使得$\|u_n\|_{Y} \rightarrow \infty$, $n\rightarrow \infty$. 可以假设$w_n\in \overline{B}(\rho_0, Y^*) $$\|u_n\|_{Y}>n , 对每个 n\in\mathbb{N} . 由(2.6)式, 存在函数 d:{{\Bbb R}} ^+\rightarrow {{\Bbb R}} ^+ 和常数 l>0 使得对每个 \|u\|_{Y}>l , 有 n 充分大使得 d(n)>\frac{\|w\|_{Y^*}\|v^*\|_{Y}}{n}+\|w\|_{Y^*} , 得到 这与假设矛盾. 则解集SOL (K, w+G(\cdot), \varphi) 是有界的, 即: 对所有的 u\in SOL (K, w+G(\cdot), \varphi) 存在常数 L_r>0 使得 \|u\|_{Y}\leq L_r .证毕. g:J\times X\rightarrow Y^* , 定义集值映射 U:J\times X\rightarrow {\cal P}(K) $$U(t, x):=\{u\in K :u\in {\rm SOL}(K, g(t, x)+G(\cdot), \varphi)\}.$$ 由定理2.1, 对于每个 (t, x)\in J\times X, U(t, x) 是非空有界闭凸的, 即: 对所有的 (t, x)\in J\times X$$ U(t, x)\in {\cal P}_{fc}(Y)$. 同时, 我们得到如下结果.

(U$_1$) $U $$J\times E$$ K$是u.s.c.;

(U$_2$) $U$是叠合可测的;

(U$_3$)$C(J, X)$中任意有界集$\Omega$, 存在$\gamma\in L^2(J, {{\Bbb R}} ^+), c>0$使得

$$$\|U(t, x)\|:=\sup\{\|u\|_{Y}:u\in U(t, x)\}\leq \gamma(t)+c\|x\|_{X}\quad\mbox{a.e. }t\in J.$$$

## 3 微分变分不等式解的存在性

(A) 算子$A$是强连续半群$T(t)$, $t\geq0$在空间$X$中的无穷小生成子, 并存在常数$M\geq1$使得$\sup\limits_{t\in[0, \infty)}\|T(t)\|\leq M$. 对任意的$t>0$, $T(t)$是紧的;

(B) 泛函$B:J\times X\rightarrow L(Y, X)$使得

(B$_1) $$t\rightarrow B(t, x)u 是可测的, 对任意的 (x, u)\in X\times Y ; (B _2)$$ x\rightarrow B^*(t, x)h$是连续的, 对所有的$h\in X$, 其中$B^*(t, x)$是算子$B(t, x)$的共轭算子;

(B$_3)$对所有的$x\in X$, 存在常数$l\geq0$使得$\|B(t, x)\|_{L(Y, X)}\leq l$;

(B$_4)$对所有的$x, y\in C(J, X) $$u\in U(t, x) , 存在函数 m(u)\in L^1(J, {{\Bbb R}} ^+) 使得 (g) g:J\times X\rightarrow Y^* 是连续和有界的, 即：存在常数 M_g>0 使得 (U _4) 集值映射 U:J\times X\rightarrow {\cal P}_{bf}(Y) 使得 h(U(t, x), U(t, y))\leq k_1(t)\|x-y\|_{X} a.e. J$$ k_1\in L^{1}(J, {\mathbb R}^+)$.

若$x$是系统(1.1), (1.2)的解, 则存在$u(t)\in U(t, x(t))$ a.e. $t\in J$, 使得

$W(t)=\|x(t)\|_{X}$, 则

对任意的$\varepsilon>0$, 考虑集值映射$\digamma:C(J, X)\rightarrow {\cal P}(C(J, X))$如下

$\int_{0}^{t-\epsilon}T(t-s-\epsilon)B(s, x(s))u(s){\rm d}s$的有界性和半群$T(t)\ (t>0)$的紧性, 得到集合

$X$中是相对紧的, 对每个$\epsilon\in(0, t) $$\delta>0 . 另外, 有 则当 \epsilon\rightarrow 0$$ \delta\rightarrow 0$时, 最后一个不等式趋向于0. 因此, 对于$\Pi(t)\; (t>0)$中任意闭集是相对紧集. 所以集合$\Pi(t)\; (t>0) $$X 中是相对紧的. 步骤5 \digamma 有闭图像. x_{n}\rightarrow x_{*}$$ C(J, X)$, $\xi_{n}\in \digamma(x_{n}) $$\xi_{n}\rightarrow \xi_{*}$$ C(J, X)$. 将证明$\xi_{*}\in \digamma(x_{*}).$这里, $\xi_{n}\in \digamma(x_{n})$表示存在$u_{n}\in U(t, x_{n})$使得

$\begin{eqnarray} \xi_{n}(t)= T(t)x_{0}+\int_{0}^{t}T(t-s)B(s, x_n(s))u_n(s){\rm d}s. \end{eqnarray}$

$$$u_{n}\rightharpoonup u_{*}, \; \; \mbox{ 对于某些 }\; u_{*}\in L^{2}(J, Y),$$$

$\begin{eqnarray} \xi_{*}(t)= T(t)x_{0}+\int_{0}^{t}T(t-s)B(s, x_*(s))u_*(s){\rm d}s. \end{eqnarray}$

## 4 微分变分不等式端点解的存在性

${\rm pr_L}:X\rightarrow X $$L -半径压缩的, 即 这映射是Lipschitz连续的. 定义 U_1(t, x)=U(t, {\rm pr_L}x) . 容易验证 U_1 满足 (U_2)$$ (U_2)$. 另外, 由${\rm pr_L}$的性质, 对a.e. $t\in J$, 所有的$x\in X $$u\in U_1(t, x) 使得 因此, 引理3.1对于 U(t, x)$$ U_1(t, x)$代替也是成立的. 因此, 不失一般性可以设对a.e. $t\in J$和所有的$x\in X$,

$$$\sup\{\|v\|_{Y}:v\in U(t, x)\}\leq\xi(t)=\gamma(t)+cL, \mbox{ 及 }\ \xi\in L^{2}(J, {{\Bbb R}} ^+).$$$

$\lambda(\cdot)\in L^2(J, \mathbb{R^+})$

$$$\left\{\begin{array}{l} x'(t)=Ax(t)+B(t, x(t))u(t), \, \, t\in J=[0, b], \\ x(0)=x_0. \end{array}\right.$$$

设$x_n\rightarrow x $$C(J, X) 中, 和 u_n\rightarrow u$$ \omega-L^2(J, Y)$. 由条件(B$_2)$和(B$_3)$, 对所有固定的$h\in C(J, X)$, 有

考虑集合

$$$Q_{\lambda}=\{u\in L^2(J, Y):\|u(t)\|_{L^2(J, Y)}\leq \lambda(t), \quad\mbox{a.e. }t\in J\},$$$

$$$g(x)(t)\in \mbox{ext}U(t, x(t)), \quad\mbox{a.e.}\quad t\in J.$$$

$$$\bigcap\limits_{n=1}^{\infty}\overline{\mbox{co}} \Big(\bigcup\limits_{k=n}^{\infty} U(t, x_k(t))\Big) \subseteq U(t, x(t))\, \, \mbox{a.e.}\, \, t\in J.$$$

设$(x_*, u_*)\in{\cal R}_{SOL}(x_0)$, 则存在$f_*\in Q_{\lambda}$使得$x_*=S(f_*) $$f_*(t)=B(t, x_*(t))u_*(t) , 其中 u_*\in S^{2}_{U}(x) 对固定的 n\geq1 , 由引理2.1, 得到对每个 x\in{\cal R}(x_0)\subseteq C(J, X) , 存在 u\in S^{2}_{U}(x) 使得对a.e. t\in J , $$\|u_*(t)-u(t)\|_{Y}\leq\frac{1}{2n}+d(u_*(t), U(t, x(t))) \leq\frac{1}{2n}+k_1(t)\|x_*(t)-x(t)\|_{X}.$$ 考虑不等式(5.1), 由文献[15, 性质2.3和定理3.1], 存在连续函数 h_n:{\cal R}(x_0)\subseteq C(J, X)\rightarrow L^1(J, X) 使得对于a.e. t\in J , 有 $$\left\{\begin{array}{l} h_n(x)(t)\in U(t, x(t)), \\ { } \|u_*(t)-h_n(x)(t)\|_{Y}\leq\frac{1}{n}+k_1(t)\|x_*(t)-x(t)\|_{X}, \end{array}\right.$$ 对每个 x\in{\cal R}(x_0) 成立. 由于 h_n(x)\in Q_{\mu}^U , 对每个 x\in{\cal R}(x_0) , 则 \|h_n(x)(t)\|_{Y}\leq\mu(t) , a.e. t\in J . 从而集合 \{h_n(x):x\in{\cal R}(x_0)\}\subseteq L^{2}(J, Y) 是一致有界的. 由文献[41, 性质2.4], 得到 h_n:{\cal R}(x_0)\subseteq C(J, X)\rightarrow L^{2}(J, Y) 是连续的. 又由文献[42, 性质6.10], 存在连续函数 g_n:{\cal R}(x_0)\subseteq C(J, X)\rightarrow L^{2}(J, Y) 对于每个 x\in{\cal R}(x_0) $$g_n(x)(t)\in {\rm ext}U(t, x(t)),$$ $$\|h_n(x)-g_n(x)\|_{\omega}<\frac{1}{n}.$$ 考虑算子 {\cal F}_n(f)={\cal B}(S(f))g_n(S(f)) . 由定理4.1, 知道 {\cal F}_n(f)$$ \omega-Q_{\lambda}$映射到$\omega-Q_{\lambda}$是连续的. 设$f_n$是映射${\cal F}_n$中固定的点. 结合(5.3)式, 有

$$$(x_n, u_n)\in{\cal R}_{{\rm ext}SOL}(x_0), \quad n\geq1,$$$

$$$u_n(t)=g_n(x_n)(t), \qquad x_n=S(f_n), \qquad f_n(t)=B(t, x_n(t))u_n(t).$$$

$H(b) $$b:T\times \Omega\times{{\Bbb R}} \rightarrow {{\Bbb R}} 是非线性泛函使得 (i) (t, z)\rightarrow b(t, z, w) 是可测的在 T\times\Omega 上对所有的 w\in {{\Bbb R}} ; (ii) w\rightarrow b^*(t, z, w)h 是连续的对所有的 h\in L^2(\Omega) , 其中 b^*(t, z, w)$$ b(t, z, w)$自共轭算子;

(iii) 对任意的$w\in L^2(\Omega),$存在正常数$l_b$使得$|b(t, z, w)|\leq l_b$对a.e. $(t, z)\in T\times \Omega$;

(iv) 对任意的$w_1, w_2\in C(T, X) $$u\in SOL(Q, \widetilde{g}(t, w(t))+\widetilde{G} (\cdot), \widetilde{\varphi}) , 存在函数 \widetilde{m}(u)\in L^1(T, {{\Bbb R}} ^+) 使得 \|b(t, z, w_1)u-b(t, z, w_2)u\|_{X}\leq \widetilde{m}(u)\|w_1-w_2\|_{X}\ \mbox{a.e. }\ t\in J, z\in \Omega. H(g)$$ g:T\times \Omega\times{{\Bbb R}} \rightarrow {{\Bbb R}}$是连续和有界的, 即：存在正常数$l_g$使得$|g(t, z, w)|\leq l_g$.

$H(0)$泛函$\eta, \beta $$\alpha 满足如下性质 定义 A:D(A)\subset X\rightarrow X$$ Aw=\Delta x$, 对每个$x\in D(A)$. 显然$A$是紧半群$T(t)$, $t>0 $$X 中的无穷小生成元[35]并且其表达式如下 w(t)(z)=w(t, z), u(t)(z)=u(t, z), (t, z)\in T\times \Omega . 考虑如下算子 \begin{eqnarray} &&(\widetilde{B}(t, w)u)(z)=b(t, z, w)u(t, z), \quad w\in L^2(\Omega), u\in W^{1, 2}(\Omega), \end{eqnarray} \begin{eqnarray} &&\widetilde{g}(t, w)(z)=g(t, z, w), \quad w\in L^2(\Omega). \end{eqnarray} 同时, 考虑泛函 \widetilde{G}:Y\rightarrow Y^*$$ \widetilde{\varphi}:Y\times Y\rightarrow {{\Bbb R}}$如下

$\begin{eqnarray} \langle \widetilde{G}(u), v\rangle&=&\int_\Omega\alpha(z)(\nabla u(z), \nabla v(z))_{{{\Bbb R}} ^N}{\rm d}z+\int_\Omega\beta(z)u(z)v(z){\rm d}z , \end{eqnarray}$

$\begin{eqnarray} \widetilde{\varphi}(u, v)&=&\int_{\Gamma_2}\eta(z)u(z){\rm d}z-\int_{\Gamma_2}\eta(z)v(z){\rm d}z , \end{eqnarray}$

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