Abstract
In this chapter, two directions for investigation are combined. On the one hand, mixed constraints are added to the formulation of the problem, that is, joint constraints on the state variable and on the control variable. Such constraints are in demand in engineering applications. On the other hand, a new and broader impulsive extension concept is considered, as it is assumed that the matrix-multiplier G may now depend on both the state variable x and the control of the conventional type u. This leads to a new, more general type of impulsive control which can be found in various engineering applications, for example, those in which rapid variations in the mass distribution of a mechanical system need to be taken into account for the small time interval when the impulse takes place. A corresponding model example of such a control system equipped with the mixed constraints is given in Sect. 6.2. Further on in this chapter, the maximum principle is proved which requires some effort and auxiliary techniques contained in Sect. 6.4. The chapter ends with ten exercises.
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Notes
- 1.
In the literature, the modulus of surjection is introduced for set-valued maps \(F:X\rightarrow 2^Y\). If spaces X and Y are finite dimensional, then
$$ \text {sur}F(x|y)=\inf \{|x^*|:\, x^*\in D^*F(x,y)(y^*),\, |y^*|=1\}. $$Here, \(D^*F(x,y)\) is the limiting coderivative of F at (x, y), [16]. By definition, \(\text {sur}F(x|y)=\infty \) if \(y\notin F(x)\). If we set \(F(\cdot ):=r(x,\cdot ,t)-C\), then \(\text {sur}M(x,u,t)=\text {sur}F(x,u,t|0)\).
- 2.
When \(\alpha \rightarrow L\), the number c approaches the number \(\displaystyle {\text {sur}M(x_*|y_*)}^{-1}\) which is called the modulus of regularity and designated by \(\text {reg}M(x_*|y_*)\). It follows that the modulus of regularity is the lower bound of all such c, for which the estimate of metric regularity still holds true; see [16].
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Arutyunov, A., Karamzin, D., Lobo Pereira, F. (2019). Impulsive Control Problems with Mixed Constraints. In: Optimal Impulsive Control. Lecture Notes in Control and Information Sciences, vol 477. Springer, Cham. https://doi.org/10.1007/978-3-030-02260-0_6
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