Guaranteed a posteriori estimation of unknown right-hand sides of linear periodic systems of ODEs

Abstract

We consider the problem of recovery of unknown right-hand sides of the first-order linear systems of ordinary differential equations with periodic coefficients from indirect observations of their solutions on a finite system of points and intervals. Under the assumption that right-hand sides and deterministic errors in observations are subjected to some quadratic restrictions, we obtain a posteriori estimates of right-hand sides which are compatible with observations data.

Keywords:

• A guaranteed a posteriori estimate
• observations
• linear periodic systems of ordinary differential equations

AMS CLASSIFICATIONS:

• 34C25
• 34B05
• 34K29

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Here and in what follows we assume that if a function $\phi (t)$ is piecewise continuous and has at a certain point $t_0$ a discontinuity of the first kind, we denote by $\mathrm{\Delta }\phi (t){|}_{t=t_0}=\phi (t_0+0)-\phi (t_0-0)$ the jump of function $\phi (t)$ at this point.

2 Let us verify, for example, (19(19) $\begin{array}{r}(C_i^{(2)}{)}^{\ast }v={\mathrm{\Phi }}^{\ast }(t_i,\cdot )v\mathrm{\forall }v\in {\mathbb{C}}^n,\end{array}$(19) ). We have $\begin{array}{rl}(C_i^{(2)}f,v{)}_n& =(Cf(t_i),v{)}_n=({\int }_0^T\mathrm{\Phi }(t_i,s)f(s)\mathrm{d}s,v{)}_n={\int }_0^T(f(s),{\mathrm{\Phi }}^{\ast }(t_i,s)v{)}_m\mathrm{d}s\\ & =(f,(C_i^{(2)}{)}^{\ast }v{)}_{\mathcal{H}}\mathrm{\forall }v\in {\mathbb{C}}^n,\end{array}$that proves (19(19) $\begin{array}{r}(C_i^{(2)}{)}^{\ast }v={\mathrm{\Phi }}^{\ast }(t_i,\cdot )v\mathrm{\forall }v\in {\mathbb{C}}^n,\end{array}$(19) ).

3 It is easy to see that the unique solvability of problems (23(22) $\begin{array}{r}{L}^{\ast }p(t)=\sum _{j=1}^M{\chi }_{{\mathrm{\Omega }}_j}(t)(H_j^{(1)}{)}^{\ast }(t)D_j^{(1)}(t)[y_j^{(1)}(t)-H_j^{(1)}(t)\hat{x}(t)]\text{for a.e.}t\in (0,T),\end{array}$(22) ) and (24(23) $\begin{array}{r}\mathrm{\Delta }p(t){|}_{t=t_i}=-(H_i^{(2)}{)}^{\ast }{D}_i^{(2)}[y_i^{(2)}-H_i^{(2)}\hat{x}(t_i)],i=1,\dots ,N,p(T)=p(0),\end{array}$(23) ) follows from the unique solvability of problems (1(1) $\begin{array}{rl}\frac{\mathrm{d}x(t)}{\mathrm{d}t}& =A(t)x(t)+B(t)f(t)\text{in}(0,T),\end{array}$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(T),\end{array}$(2) ) since normalized fundamental matrix $Z(t)$ of the system $-\frac{\mathrm{d}p(t)}{\mathrm{d}t}=A^{\ast }p(t)$adjoint to (3(3) $\begin{array}{r}\frac{\mathrm{d}x(t)}{\mathrm{d}t}=A(t)x(t)\end{array}$(3) ) satisfies the condition (see [8,9]) (1) $\begin{array}{r}det(E_n-Z(T))\ne 0.\end{array}$(1) It is known that $Z(t)=[X^{\ast }(t){]}^{-1}$.

4 In fact, by virtue of generalized Cauchy–Schwarz inequality [12, p.186], $\underset{\tilde{f}\in {\tilde{G}}_y}{sup}|\mathrm{\Re }(a,\tilde{f}{)}_{\mathcal{H}}|\le \underset{\tilde{f}\in {\tilde{G}}_y}{sup}|(a,\tilde{f}{)}_{\mathcal{H}}|\le \underset{\tilde{f}\in {\tilde{G}}_y}{sup}({\tilde{Q}}^{-1}a,a{)}_{\mathcal{H}}^{1/2}(\tilde{Q}\tilde{f},\tilde{f}{)}_{\mathcal{H}}^{1/2}={\gamma }^{1/2}({\tilde{Q}}^{-1}a,a{)}_{\mathcal{H}}^{1/2}$and this inequality is transformed to an equality on the element $\tilde{f}={\gamma }^{1/2}({\tilde{Q}}^{-1}a,a{)}_{\mathcal{H}}^{-1/2}{\tilde{Q}}^{-1}a.$