Abstract
In this chapter, we consider differentiability properties involving the natural state. We present three results. These results depend on the differentiability of the input–output system. The first gives the differentiability of the natural state. The second gives the differentiability of the map from past input to natural state. The last result implies the differentiability of the natural state trajectories. The result also provides a differential equation representation for the natural state trajectories.
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Notes
- 1.
The space U is shift continuous if \(\lim _{h\rightarrow 0}\left \Vert u -\mathrm{ L}_{h}u\right \Vert _{s,t}\) = 0 for all u for which the norm is defined and for all −∞ ≤ s < t < ∞.
- 2.
This example is a continuation of Example 1 [14].
- 3.
From the definition of a symmetrized kernel, \(f_{\mathrm{sym}}\left (r,\tau _{1},\tau _{2})\right ) = \frac{1} {2}\left (f(r,\tau _{1},\tau _{2}) + f(r,\tau _{2},\tau _{1})\right )\). Hence, the symmetry property, \(f_{\mathrm{sym}}\left (r,\tau _{1},\tau _{2}\right )\) \(= f_{\mathrm{sym}}\left (r,\tau _{2},\tau _{1}\right )\). To show that using the regular (unsymmetrized) or symmetrized kernel within the integral (4.12) is the same, we have \(\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f_{\mathrm{sym}}\left (r,\tau _{1},\tau _{2}\right )u\left (r -\tau _{1}\right )u\left (r -\tau _{2}\right )d\tau _{1}d\tau _{2} =\int _{ -\infty }^{\infty }\int _{-\infty }^{\infty }\frac{1} {2}\left (f\left (r,\tau _{1},\tau _{2}\right ) + f\left (r,\tau _{2},\tau _{1}\right )\right )u\left (r -\tau _{1}\right )u\left (r -\tau _{2}\right )d\tau _{1}d\tau _{2} = \frac{1} {2}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(r,\tau _{1},\tau _{2})u\left (r -\tau _{1}\right )u\left (r -\tau _{2}\right )d\tau _{1}d\tau _{2}\quad + \frac{1} {2}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(r,\tau _{2},\tau _{1})u\left (r -\tau _{2}\right )u\left (r -\tau _{1}\right )d\tau _{2}d\tau _{1} =\int _{ -\infty }^{\infty }\int _{-\infty }^{\infty }f\left (r,\tau _{1},\tau _{2}\right )u\left (r -\tau _{1}\right )u\left (r -\tau _{2}\right )d\tau _{1}d\tau _{2}\).
- 4.
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© 2014 Demetrios Serakos
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Serakos, D. (2014). State Differentiability Properties in Input–Output Systems. In: State Space Consistency and Differentiability. SpringerBriefs in Optimization. Springer, Cham. https://doi.org/10.1007/978-3-319-14469-6_4
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DOI: https://doi.org/10.1007/978-3-319-14469-6_4
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