Variational calculus

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Problem: For the Bolza problem find the admissible extremal and determine its type: \[ B=\int_{1}^{2} t^{2} \dot{x}^{2} d t-2 x(1)+x^{2}(2) \rightarrow \text { extr. } \]

4.16 Variational calculus

3.85 $

Problem: Find the eigenvalues and the eigenfunctions of the functional: \[ \begin{array}{l} \mathcal{T}[y]=\int_{0}^{1}\left(y^{2}+\left(y^{\prime}\right)^{2}\right) d x, \quad y(0)=y(1)=0, \\ \int_{0}^{1} y^{2} d x=0 . \end{array} \]

4.17 Variational calculus

3.85 $

Problem: Solve the Bolza problem: \[ B(x(\cdot))=\int_{1}^{2} t^{2} \dot{x}^{2} d t-2 x(1)+x^{2}(2) \rightarrow \text { extr. } \]

4.18 Variational calculus

4.37 $

Problem: For the simplest problem of the classical calculus of variations describe the set of admissible functions and investigate the functional for an extremum. \[ \begin{array}{l} \mathrm{J}=\int_{1}^{2}\left(\dot{x}^{2}-t^{2} x\right) d t \rightarrow \text { extr }, \\ x(0)=0, \quad x(1)=0 . \end{array} \]

4.19 Variational calculus

4.37 $

Problem: Solve the problem of finding the extrema of the functional reducing it to an optimal control problem with restrictions on the phase coordinates and controls: \[ \begin{array}{l} \int_{0}^{2} x d t \rightarrow \text { extr }, \quad|\ddot{x}| \leq 2, \quad x(0)+x(2)=0, \\ \dot{x}(0)=0 . \end{array} \]

4.20 Variational calculus

6.42 $

Problem: Solve the problem with moving boundaries: \[ \begin{array}{l} J(x(\cdot))=\int_{0}^{T_{0}}\left(\dot{x}^{2}-x\right) d t \rightarrow \text { extr }, \quad x(0)=0, \\ x\left(T_{0}\right)=\xi . \end{array} \]

4.21 Variational calculus

5.14 $

Problem: Solve the problem of Bolza: \[ \int_{1}^{\mathrm{e}} 2 \dot{x}(t \dot{x}+x) d t+3 x^{2}(1)-x^{2}(e)-4 x(e) \rightarrow \text { extr. } \]

4.22 Variational calculus

6.42 $

Problem: Solve the simplest problem of the classical calculus of variations: \[ \begin{array}{l} \int_{0}^{T_{0}}\left(2 \dot{x}^{2}+x^{2}-4 x \sin t\right) d t \rightarrow \text { extr }, \\ x(0)=0, \quad x\left(T_{0}\right)=\xi . \end{array} \]

4.23 Variational calculus

5.14 $

Problem: Solve the problem with moving boundaries: \[ \begin{array}{l} \int_{0}^{1}\left(\frac{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}}{2}-x_{1} x_{2}\right) d t \rightarrow \text { extr }, \\ x_{1}(0)=x_{2}(0)=1 \end{array} \]

4.24 Variational calculus

3.85 $

Problem: Solve the problem with moving boundaries: \[ \begin{array}{l} \int_{0}^{T_{0}}\left(\ddot{x}^{2}+\dot{x}^{2}\right) d t \rightarrow \text { extr, } \\ x(0)=\dot{x}(0)=x\left(T_{0}\right)=\dot{x}\left(T_{0}\right)=0 . \end{array} \]

4.25 Variational calculus

5.14 $

Problem: Solve the optimal control problem: \[ \begin{array}{l} \int_{0}^{1}\left(x^{2}+u^{2}\right) d t \rightarrow \text { extr }, \quad \dot{x}=x+u, \\ x(1)=1 . \end{array} \]

4.26 Variational calculus

6.42 $

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