Abstract

The questions of weakly generalized solvability of a nonlinear inverse problem in nonlinear optimal control of thermal processes for a parabolic differential equation are studied. The parabolic equation is considered under initial and boundary conditions. To determine the recovery function, a nonlocal integral condition is specified. Moreover, the recovery function nonlinearly enters into the differential equation. Is applied the method of variable separation based on the search for a solution to the mixed inverse problem in the form of a Fourier series. It is assumed that the recovery function and nonlinear term of the given differential equation are also expressed as a Fourier series. For fixed values of the control function, the unique solvability of the inverse problem is proved by the method of compressive mappings. The quality functional has a nonlinear form. The necessary optimality conditions for nonlinear control are formulated. The determination of the optimal control function is reduced to a complicated functional-integral equation, the process of solving which consists of solving separately taken two nonlinear functional and nonlinear integral equations. Nonlinear functional and integral equations are solved by the method of successive approximations. Formulas are obtained for the approximate calculation of the state function of the controlled process, the recovery function, and the optimal control function. Is proved the absolutely and uniformly convergence of the obtained Fourier series.

中文翻译：

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非线性逆问题中热过程的非线性最优控制

摘要

研究了抛物型微分方程热过程的非线性最优控制中的非线性逆问题的弱广义可解性问题。在初始和边界条件下考虑抛物线方程。为了确定恢复函数，指定了非局部积分条件。此外，恢复函数非线性地进入微分方程。应用了基于傅立叶级数形式的混合逆问题的解的变量分离方法。假定给定微分方程的恢复函数和非线性项也表示为傅立叶级数。对于控制函数的固定值，通过压缩映射的方法证明了反问题的独特可解性。质量函数具有非线性形式。制定了非线性控制的必要最优条件。最优控制函数的确定被简化为一个复杂的函数积分方程，求解过程包括分别求解两个非线性泛函和非线性积分方程。非线性泛函和积分方程通过逐次逼近法求解。获得用于对受控过程的状态函数，恢复函数和最佳控制函数进行近似计算的公式。证明了所获得傅里叶级数的绝对一致收敛。最优控制函数的确定简化为一个复杂的函数积分方程，求解过程包括分别求解两个非线性函数方程和非线性积分方程。非线性泛函和积分方程通过逐次逼近法求解。获得用于对受控过程的状态函数，恢复函数和最佳控制函数进行近似计算的公式。证明了所获得傅里叶级数的绝对一致收敛。最优控制函数的确定被简化为一个复杂的函数积分方程，求解过程包括分别求解两个非线性泛函和非线性积分方程。非线性泛函和积分方程通过逐次逼近法求解。获得用于对受控过程的状态函数，恢复函数和最佳控制函数进行近似计算的公式。证明了所获得傅里叶级数的绝对一致收敛。获得用于对受控过程的状态函数，恢复函数和最佳控制函数进行近似计算的公式。证明了所获得傅里叶级数的绝对一致收敛。获得用于对受控过程的状态函数，恢复函数和最佳控制函数进行近似计算的公式。证明了所获得傅里叶级数的绝对一致收敛。