Recent results on stability of periodic linear differential equations (Q2912632)

This short note is a summary of results published elsewhere.NEWLINENEWLINEThe first part deals with the stability, in the sense of Lyapunov, of the solutions of the Hill equation NEWLINE\[NEWLINE u''(t) + a(t) u(t) = 0,\quad t\in \mathbb{R}, NEWLINE\]NEWLINE where \(a\in L^1(0,T)\) is \(T\)-periodic. This stability is related to the eigenvalues of the boundary problem associated to the equation \(u''(t) + (\lambda+a(t)) u(t) = 0\) with periodic and antiperiodic boundary conditions.NEWLINENEWLINEIn Theorem~1, whose proof can be found in [the authors, J. Math.\ Anal.\ Appl.\ 376, No. 2, 429--442 (2011; Zbl 1222.34017)], the authors claim that if there exist \(p\in \mathbb{N}\) and \( k\in [p^2\pi^2/T^2,(p+1)^2\pi^2/T^2]\) such that \(k\leq a\) and NEWLINE\[NEWLINE \|a\|_{L^1(0,T)} \leq kT + k^{1/2} 2 (p+1) \cot \frac{k^{1/2}T}{2(p+1)}, NEWLINE\]NEWLINE then \(\lambda = 0\) is in the stability zone of order~\(n\) of the Hill equation.NEWLINENEWLINEThe second part is devoted to give sufficient conditions to ensure that the time periodic system NEWLINE\[NEWLINE u''(t)+P(t) u(t) = 0, \quad t \in \mathbb{R}, NEWLINE\]NEWLINE where \(P\) is a real \(n\times n\) matrix with continuous coefficients such that the system has no non-trivial constant solutions and such that NEWLINE\[NEWLINE \int_0^T \langle P(t)k,k\rangle \, dt \geq 0, \quad \forall k \in \mathbb{R}^n, NEWLINE\]NEWLINE is stablely bounded, that is, that the solutions of any nearby system of the same type are bounded.NEWLINENEWLINETheir result, proven in [the authors, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 5, 1913--1925 (2011; Zbl 1229.34010)], states that if there exists a diagonal matrix \(B(t)\), with \(T\)-periodic continuous coefficients \(b_{ii}(t)\), such that \(B(t)-P(t)\) is semidefinite positive and \(p_i\in [1,\infty]\) such that the \(L_{p_i}\) norms of the positive part of the coefficients \(b_{ii}\) are bounded by appropriate constants related to the boundary problem with antiperiodic boundary conditions, then the system is stablely bounded.NEWLINENEWLINEFor the entire collection see [Zbl 1243.00023].