SIAM Journal on Control and Optimization, Volume 61, Issue 6, Page 3664-3694, December 2023.
Abstract. This article focuses on the controllability for a class of semilinear evolution equations and applications to some specific differential equations. Without assuming compactness or equicontinuity of the related semigroups, the existence of mild solutions of semilinear equations in Hilbert space is demonstrated via the noncompact measure tool and the fixed point trick. In order to study the controllability of semilinear equations, new concepts are proposed, namely, exact controllability along any [math]-bounded Lipschitz continuous curve, and approximate controllability along any continuous curve. Furthermore, two new approximation methods, “bisection method” and “equisection method,” are introduced. The exact controllability along any [math]-bounded Lipschitz continuous curve and the approximate controllability along any continuous curve in the sense of the graph norm of semilinear evolution equations are obtained under the asymptotic condition on the norm of the controllability Gramian inverse operator near the zero point. In fact, our conclusions show that this is not only a result control but also a process control. Finally, the results presented in this paper are employed in resistance, inductance, voltage source type electrical circuit systems, one-dimensional nonhomogeneous transport systems, as well as differential equations with delay which have important effects on economic systems.

中文翻译：

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半线性演化方程的过程可控性及其应用

SIAM 控制与优化杂志，第 61 卷，第 6 期，第 3664-3694 页，2023 年 12 月。
摘要。本文重点讨论一类半线性演化方程的可控性及其在一些特定微分方程中的应用。在不假设相关半群紧致或等连续的情况下，通过非紧测度工具和不动点技巧证明了希尔伯特空间中半线性方程温和解的存在性。为了研究半线性方程的可控性，提出了新的概念，即沿任意有界Lipschitz连续曲线的精确可控性和沿任意连续曲线的近似可控性。此外，还引入了两种新的近似方法：“二分法”和“等分法”。在可控性格拉米亚逆算子在零附近范数的渐近条件下，得到了半线性演化方程图范数意义上沿任意有界Lipschitz连续曲线的精确可控性和沿任意连续曲线的近似可控性观点。事实上，我们的结论表明，这不仅是结果控制，也是过程控制。最后，本文的结果应用于电阻、电感、电压源型电路系统、一维非齐次输运系统以及对经济系统有重要影响的时滞微分方程。