Null-controllability of the Kolmogorov equation in the whole phase space

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• 作者：Jé ; rô ; me Le Rousseau^a ; ^{jlr@univ-orleans.fr" class="auth_mail" title="E-mail the corresponding author} ; Ivá ; n Moyano^b ; ^{ivan.moyano@math.polytechnique.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Controllability ; Unbounded domain ; Carleman estimates ; Spectral inequality
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：260
• 期：4
• 页码：3193-3233
• 全文大小：575 K

文摘

We prove the null controllability, in arbitrary positive time, of the Kolmogorov equation class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005367&_mathId=si1.gif&_user=111111111&_pii=S0022039615005367&_rdoc=1&_issn=00220396&md5=38fb84310346ea9dc9eb09ca6b00dbbb" title="Click to view the MathML source">∂_t+v⋅∇_x−△_vclass="mathContainer hidden">class="mathCode">${\partial }_t+v\cdot {\mathrm{\nabla }}_x-{△}_v$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005367&_mathId=si2.gif&_user=111111111&_pii=S0022039615005367&_rdoc=1&_issn=00220396&md5=a415c86c877029c77c8aeb92323e2bc6" title="Click to view the MathML source">(x,v)∈R^d×R^dclass="mathContainer hidden">class="mathCode">$(x,v)\in {\mathbb{R}}^d\times {\mathbb{R}}^d$, with a control region of the form class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005367&_mathId=si3.gif&_user=111111111&_pii=S0022039615005367&_rdoc=1&_issn=00220396&md5=66827e24037acd114accdc6b1a9e1c0a" title="Click to view the MathML source">ω=ω_x×ω_vclass="mathContainer hidden">class="mathCode">$\omega ={\omega }_x\times {\omega }_v$, where both class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005367&_mathId=si29.gif&_user=111111111&_pii=S0022039615005367&_rdoc=1&_issn=00220396&md5=c3f2942055c46696fc2b10f08ee4eb58" title="Click to view the MathML source">ω_xclass="mathContainer hidden">class="mathCode">${\omega }_x$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005367&_mathId=si30.gif&_user=111111111&_pii=S0022039615005367&_rdoc=1&_issn=00220396&md5=f0eda2a1b1873911fe151376b2ac7548" title="Click to view the MathML source">ω_vclass="mathContainer hidden">class="mathCode">${\omega }_v$ are open subsets of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005367&_mathId=si20.gif&_user=111111111&_pii=S0022039615005367&_rdoc=1&_issn=00220396&md5=cae5d6830d41d62e4292fc555d981b61" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$ that are sufficiently spread out throughout the whole space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005367&_mathId=si20.gif&_user=111111111&_pii=S0022039615005367&_rdoc=1&_issn=00220396&md5=cae5d6830d41d62e4292fc555d981b61" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$. The proof is based on, on the one hand, a spectral inequality in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005367&_mathId=si20.gif&_user=111111111&_pii=S0022039615005367&_rdoc=1&_issn=00220396&md5=cae5d6830d41d62e4292fc555d981b61" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$ with an observation on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005367&_mathId=si29.gif&_user=111111111&_pii=S0022039615005367&_rdoc=1&_issn=00220396&md5=c3f2942055c46696fc2b10f08ee4eb58" title="Click to view the MathML source">ω_xclass="mathContainer hidden">class="mathCode">${\omega }_x$, and, on the other hand, a Carleman-based observability inequality for a family of parabolic operators, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615005367&_mathId=si7.gif&_user=111111111&_pii=S0022039615005367&_rdoc=1&_issn=00220396&md5=eead45d7f2bfb811d741b082cfaef2fc" title="Click to view the MathML source">∂_t−iv⋅ξ−△_vclass="mathContainer hidden">class="mathCode">${\partial }_t-iv\cdot \xi -{△}_v$, coupled with a knowledge of the decay rate of the free solutions of the Kolmogorov equation.