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Additional resources for A Basic Theorem in the Computation of Ellipsoidal Error Bounds

Sample text

To quantify the error between the original and approximating systems, we first define the size of the discretization of The diameter of the box is where is the Euclidean norm. Then, the size of the discretization of is Lemma 2. Let be the size of the discretization of and be the Lipschitz constant of the function in (1). Let be a solution of (1). Then, for all where is the Haussdorf distance. 36 E. Asarin and T. Dang The above lemma gives a bound on the distance between the derivatives of the original system and the approximating system.

Therefore, using (15) to compute reachable states we only need to compute the matrix exponential Lemma 3. Let be a solution of (4) under a fixed input and the approximate solution obtained by the scheme (15) with the same input such that If the derivative is bounded by then for all be The proof of the lemma uses standard results on the remainder term of Hermite interpolating polynomials , and it is omitted here. The lemma shows that the error of the scheme (15) is of order As shown earlier, the error due to the restriction to piecewise constant inputs is quadratic; hence, this additional error does not change the order of the method.

E. e. state To this end, we need a way to discern between the paths using the measurements they produce through As we are about to show, the only way to achieve that without any information other than the available measurement is by taking advantage of the following inclusion, immediate from where denotes the column range space of the matrix M. The question is then whether which would provide us with a simple procedure for recovering a path from the measurements (using the range inclusion test, see Appendix A): The main issue lies in whether the test (10) has a unique solution. 