本文選題：反應(yīng)-擴(kuò)散方程組 + 零能控性　； 參考：《哈爾濱工業(yè)大學(xué)》2015年碩士論文

【摘要】：自然界中的很多現(xiàn)象都可以用反應(yīng)擴(kuò)散方程組來描述,這些方程組的能控性研究也顯得越來越重要,本文研究了一類含梯度項(xiàng)的非線性反應(yīng)擴(kuò)散方程組在內(nèi)部一個任意小的區(qū)域上施加控制函數(shù)時的零能控性。本文首先考慮該非線性方程組所對應(yīng)的線性方程組,給出了這類線性方程組解的存在性、唯一性和正則性。在此基礎(chǔ)上構(gòu)造相應(yīng)的泛函,利用該泛函的極小元來給出控制函數(shù),而泛函極小元的存在性依賴于該線性方程組的對偶方程組的能觀性估計。在證明能觀不等式時,由于所考慮的方程組中含有梯度項(xiàng),經(jīng)典的Carleman不等式不再適用,本文對此不等式進(jìn)行改進(jìn)。最終,證明了該線性系統(tǒng)在內(nèi)部一個任意小的區(qū)域上施加兩個控制函數(shù)時是零能控的。進(jìn)一步,減少控制函數(shù)的個數(shù)是控制理論的一個重要研究部分�？梢园l(fā)現(xiàn),控制函數(shù)個數(shù)的減少可以通過改進(jìn)Carleman不等式來實(shí)現(xiàn),為此本文對該線性方程組施加一些限制條件,在方程的Carleman不等式的基礎(chǔ)上,給出了方程組的Carleman不等式,從而證明在內(nèi)部一個任意小的區(qū)域上只施加一個控制函數(shù)時該系統(tǒng)也是零能控的。另一方面,“虛擬函數(shù)法”也是一種減少控制函數(shù)個數(shù)的有效方法,本文在施加兩個控制時方程組的零能控性結(jié)論的基礎(chǔ)上,利用“虛擬函數(shù)法”,亦證明了當(dāng)該方程組滿足一定條件時,僅施加單一控制函數(shù),該線性方程組也是零能控的。最后,將原來的非線性方程組線性化,結(jié)合Schauder不動點(diǎn)定理,證明了上述有關(guān)線性方程組的零能控性結(jié)論對于相應(yīng)的非線性方程組也是成立的,從而證明了本文的主要結(jié)論。
[Abstract]:Many phenomena in nature can be described by reaction-diffusion equations, and the controllability of these equations is becoming more and more important. In this paper, we study the zero controllability of a class of nonlinear reaction-diffusion equations with gradient terms when a control function is applied to an arbitrary small region within the system. In this paper, we first consider the system of linear equations corresponding to the system of nonlinear equations, and give the existence, uniqueness and regularity of the solutions of the system of linear equations. On this basis, the corresponding functional is constructed and the control function is given by using the minimal element of the functional. The existence of the minimal element of the functional depends on the estimation of the observability of the dual equations of the system of linear equations. In order to prove the observable inequality, the classical Carleman inequality is no longer applicable because there are gradient terms in the system of equations under consideration, so the inequality is improved in this paper. Finally, it is proved that the linear system is zero controllable when two control functions are applied on an arbitrary small region within the system. Further, reducing the number of control functions is an important part of control theory. It can be found that the reduction of the number of control functions can be achieved by the improved Carleman inequality. In this paper, some restrictions are imposed on the system of linear equations, and on the basis of the Carleman inequality of the equation, the Carleman inequality of the equations is given. It is proved that the system is also zero controllable when only one control function is applied to an arbitrary small region. On the other hand, "virtual function method" is also an effective method to reduce the number of control functions. It is also proved that when the system of equations satisfies certain conditions, only a single control function is applied, and the system of linear equations is also zero controllable. Finally, by linearizing the original nonlinear equations and combining with Schauder fixed point theorem, it is proved that the zero controllability of the system of linear equations mentioned above is also true for the corresponding nonlinear equations, and the main conclusions in this paper are proved.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O175

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