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By Fredrik Ejserholm, Alexander Vastesson, Tommy Haraldsson, Wouter van der Wijngaart, Jens Schouenborg, Lars Wallman, Martin Bengtsson

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UT C! v T;t C! Rt ; d t C! 0t;1 /0;1 : For t D T , limits are taken from the right in computing these terms. 10). ut C! ut C! Rt ; d t C! 11) have the same form as the domain and range of a natural state. Example 27. 12) 0 1 < r < 1, where, purely for simplicity, the input time functions, the kernel f , and hence the output time functions are scalar valued. Obviously such a system is time-varying unless the kernel f is not actually a function of its first argument. r; 2 ; 2 / is symmetric in 1 and 2 .

Ut C! h//t C! u C! h// C! h//t C! h//t C! h//t C! h//t C! h/ks D kd k : h Therefore, since the first limit in I is zero, I is zero. utCh C! Lh u/t C! Lh u/ C! Lh u/t C! h//t C! h//t C! h//t C! h//t C! 0 : where G and H are from Definition 25. h//t C! ut C! h//t C! h//t C! h//t C! 9) the derivative of the state trajectory t ! tu is lim u tCh . 0 / h u t. ut C! ut C! Rt ; d t C! 0t;1 //0;1 ; . 10) For time-invariant systems, differentiability of the natural state trajectory follows from the general conditions and the shift differentiability of the input FF.

Of the N Š terms it gives, only the identity permutation term has its indices in the specified order: 2 1 f 1;:::;N . N / f . t 1 has a single term with the permutation f 1;:::;N . 1/ 1;:::;N . 3) is divided by N Š, we see that under the condition of all ri D 1 the symmetrization does not alter the operator. Next, suppose r1 ¤ 1 and all other ri D 1. With the indices in their specific order, the outside term is ~ 1; 1;2;:::;P f . t N/ : This term gives r1 Š inner terms with indices in this order and they are all identical.

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