In the recent paper by Cascales, Kadets, Orihuela and Wingler it is shown that for every strictly convex Banach space $X$ every non-expansive bijection $F: B_X \to B_X$ is an isometry. We extend this result to the space $\ell_1$, which is not strictly convex.
Keywords: non-expansive map; unit ball; plastic space.
2000 Mathematics Subject Classification: 46B20. [ Full-text available (PDF) ] Top of the page.
Linear matrix equations widely used in the theory of stability of motion, control theory and signal processing. We suggest an algorithm for regularization of the inhomogeneous generalized matrix equation and, in particular, the Sylvester equation in general case when the linear matrix operator $L,$ corresponding to the homogeneous part of the linear generalized matrix equation, has no inverse.
Keywords: Lyapunov matrix equation, Sylvester matrix equation, conditions of regularization, pseudoinverse matrix.
2000 Mathematics Subject Classification: 15A24, 34В15, 34C25. [ Full-text available (PDF) ] Top of the page.
The time-optimal control problem for the system $\dot x_1=u$, $\dot x_2=x_1$, $\dot x_3=x_1^3$ is considered. Explicit formulas for finding optimal controls are given. The explicit solution of the optimal synthesis problem is obtained.
Keywords: : control systems, time optimality, explicit solution.
2000 Mathematics Subject Classification: 93C10, 49K30. [ Full-text available (PDF) ] Top of the page.
Null-Controllability of any evolution partial differential equation with constant coefficients in the space of infinitely differentiable rapidly decreasing functions is proved. Conditions under which a control is independent of time are given. Bang-bang controls for the classical equations of mathematical physics are considered.
Keywords: null-controllability, boundary value problem, Fourier transform, bang-bang controls.
2000 Mathematics Subject Classification: 35S10. [ Full-text available (PDF) ] Top of the page.
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