【摘要】：分?jǐn)?shù)階微積分是整數(shù)階微積分的延伸與拓展,其發(fā)展幾乎與整數(shù)階微積分的發(fā)展同步。分?jǐn)?shù)階微積分在越來越多的領(lǐng)域中都發(fā)揮著極其重要的作用。與整數(shù)階模型相比,分?jǐn)?shù)階模型能夠更加準(zhǔn)確地描述自然現(xiàn)象,更好地模擬自然界的物理現(xiàn)象和動態(tài)過程。隨著分?jǐn)?shù)階微積分理論在不同的科學(xué)領(lǐng)域里出現(xiàn),對其理論或應(yīng)用價值的研究都顯得尤為迫切。因此,對分?jǐn)?shù)階微分方程和系統(tǒng)進(jìn)行深入研究具有廣泛的理論意義與實際應(yīng)用價值。有關(guān)分?jǐn)?shù)階微分方程和系統(tǒng)的研究引起了國內(nèi)外學(xué)者的廣泛關(guān)注并逐漸成為一個熱點問題。本文針對幾類分?jǐn)?shù)階系統(tǒng)的穩(wěn)定性分析、鎮(zhèn)定控制器設(shè)計問題和兩類分?jǐn)?shù)階微分方程邊值問題解的存在性進(jìn)行了研究,給出了幾類分?jǐn)?shù)階系統(tǒng)一些新的穩(wěn)定性判據(jù)、鎮(zhèn)定控制器的設(shè)計方法和分?jǐn)?shù)階微分方程邊值問題解存在的若干充分條件,并分別用仿真例子驗證了所得到結(jié)果的有效性。主要研究內(nèi)容如下：1.基于Caputo分?jǐn)?shù)階導(dǎo)數(shù)已有的基本性質(zhì),給出了Caputo分?jǐn)?shù)階導(dǎo)數(shù)的一些新性質(zhì)。這些新性質(zhì)可以幫助尋找一個給定的分?jǐn)?shù)階系統(tǒng)的二次Lyapunov函數(shù)。2.研究了幾類分?jǐn)?shù)階系統(tǒng)的穩(wěn)定性和鎮(zhèn)定性。首先,利用分?jǐn)?shù)階Lyapunov函數(shù)方法,研究了分?jǐn)?shù)階線性系統(tǒng)的穩(wěn)定性,并給出了分?jǐn)?shù)階線性受控系統(tǒng)的狀態(tài)反饋控制器設(shè)計。其次,利用分?jǐn)?shù)階Razumikhin定理,研究了分?jǐn)?shù)階線性時滯系統(tǒng)的穩(wěn)定性,并給出了分?jǐn)?shù)階線性時滯受控系統(tǒng)的狀態(tài)反饋控制器設(shè)計。最后,利用分?jǐn)?shù)階Lyapunov函數(shù)方法,研究了分?jǐn)?shù)階非線性系統(tǒng)的穩(wěn)定性,并利用Backstepping設(shè)計方法,給出了一類分?jǐn)?shù)階非線性三角系統(tǒng)的狀態(tài)反饋控制器設(shè)計。3.研究了分?jǐn)?shù)階非線性三角系統(tǒng)的反饋鎮(zhèn)定控制器設(shè)計問題。通過引入適當(dāng)?shù)臓顟B(tài)變換,將分?jǐn)?shù)階非線性三角系統(tǒng)的反饋鎮(zhèn)定控制器設(shè)計問題轉(zhuǎn)化為待定參數(shù)的選取問題。利用靜態(tài)增益控制設(shè)計方法和分?jǐn)?shù)階Lyapunov函數(shù)方法,分別給出了分?jǐn)?shù)階非線性下、上三角系統(tǒng)的狀態(tài)反饋和輸出反饋控制器設(shè)計。4.研究了分?jǐn)?shù)階非線性時滯三角系統(tǒng)的反饋鎮(zhèn)定控制器設(shè)計問題。通過引入適當(dāng)?shù)臓顟B(tài)變換,將分?jǐn)?shù)階非線性時滯三角系統(tǒng)的反饋鎮(zhèn)定控制器設(shè)計問題轉(zhuǎn)化為待定參數(shù)的選取問題。利用靜態(tài)增益控制設(shè)計方法和分?jǐn)?shù)階Razumikhin定理,分別設(shè)計了分?jǐn)?shù)階非線性時滯下、上三角系統(tǒng)的狀態(tài)反饋和輸出反饋控制器。5.研究了兩類分?jǐn)?shù)階微分方程邊值問題解的存在性。利用上下解方法、Shauder不動點定理和Leggett-Williams不動點定理,建立了一類分?jǐn)?shù)階微分方程邊值問題至少存在一個或三個正解的幾個充分條件。利用Banach代數(shù)上的Dhage不動點定理,給出了一類混合分?jǐn)?shù)階微分方程邊值問題存在一個解的充分條件。
[Abstract]:Fractional calculus is an extension and extension of integral order calculus, and its development almost keeps pace with the development of integer order calculus. Fractional calculus plays an important role in more and more fields. Compared with the integer order model, the fractional order model can describe the natural phenomena more accurately and simulate the physical phenomena and dynamic processes better. With the emergence of fractional calculus theory in different fields of science, it is urgent to study its theory or application value. Therefore, the study of fractional differential equations and systems has a wide range of theoretical significance and practical application value. The study of fractional differential equations and systems has attracted the attention of scholars at home and abroad and has gradually become a hot issue. In this paper, the existence of solutions to the stability analysis, stabilization controller design problem and boundary value problem of two kinds of fractional differential equations are studied, and some new stability criteria are given. The design method of stabilizing controller and some sufficient conditions for the existence of solutions to the boundary value problem of fractional differential equations are discussed. Simulation examples are given to verify the validity of the obtained results. The main research contents are as follows: 1. Based on the basic properties of Caputo fractional derivative, some new properties of Caputo fractional derivative are given. These new properties can help to find the quadratic Lyapunov function of a given fractional system. The stability and stability of several fractional order systems are studied. Firstly, the stability of fractional linear systems is studied by using the fractional Lyapunov function method, and the state feedback controller design for fractional linear controlled systems is given. Secondly, the stability of fractional linear time-delay systems is studied by using fractional Razumikhin theorem, and the state feedback controller design for fractional linear time-delay controlled systems is given. Finally, the stability of fractional nonlinear systems is studied by using the fractional Lyapunov function method, and the state feedback controller design of a class of fractional nonlinear triangular systems is given by using the Backstepping design method. The design of feedback stabilization controllers for fractional nonlinear triangular systems is studied. By introducing appropriate state transformation, the design problem of feedback stabilization controller for fractional nonlinear triangular systems is transformed into the selection of undetermined parameters. Using the static gain control design method and the fractional Lyapunov function method, the state feedback and output feedback controller design of the upper triangular system under fractional order nonlinearity are given respectively. 4. In this paper, the design of feedback stabilization controller for fractional nonlinear delay-triangular systems is studied. By introducing appropriate state transformation, the problem of feedback stabilization controller design for fractional nonlinear delay-triangular systems is transformed into the selection of undetermined parameters. Using the static gain control design method and fractional order Razumikhin theorem, the state feedback and output feedback controllers for upper triangular systems with fractional nonlinear delay are designed respectively. The existence of solutions for two kinds of boundary value problems for fractional differential equations is studied. By using the upper and lower solution method and the Leggett-Williams fixed point theorem, some sufficient conditions for the existence of at least one or three positive solutions for a class of fractional differential equation boundary value problems are established. By using the Dhage fixed point theorem on Banach algebra, a sufficient condition for the existence of a solution to the boundary value problem for a class of mixed fractional differential equations is given.
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：TP13

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