Abstract
The problem of a prescribed discrete route realization by a controlled linear system in the presence of unknown bounded disturbance is considered. The problem is solved based on an auxiliary cheap control linear-quadratic differential game. Novel solvability conditions are established. A numerical example is presented.
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Communicated by Meir Pachter.
Appendices
Appendix 1
1.1 Proof of Theorem 3.1
Due to Theorem 1 of [4], it should be proved that for a fixed \(\beta >0\) and for a sufficiently small \(\alpha >0\), the inequality (22) is valid.
First, let us establish the auxiliary propositions.
Proposition 6.1
The matrices \({\mathcal F}_u(t)\) and \({\mathcal F}_v(t)\), \(t\in [t_0, t_f]\), are symmetric and positive semi-definite.
Proof
Thus, by (9), the matrices \({\mathcal F}_u(t)\) and \({\mathcal F}_v(t)\), \(t\in [t_0, t_f]\), are symmetric.
Let \(t\in [t_0,t_f]\) be fixed and \(m_t^{(i)}\in \mathbb {R}^n\), \(i=q(t),\ldots ,K\), be arbitrary n-vectors, constituting an arbitrary vector \(m_t\in \mathbb {R}^{p(t)}\):
By direct calculation,
where
Thus, the matrices \({\mathcal F}_u(t)\) and \({\mathcal F}_v(t)\), \(t\in [t_0, t_f]\), are positive semi-definite. \(\square \)
Proposition 6.2
If Condition A holds, then
while
Proof
Assume that \(m_t\in \mathrm{Ker}{\mathcal F}_u(t)\), i.e., \({\mathcal F}_u(t)m_t=0\), providing
Note that Condition A guarantees the positive definiteness of the matrix
and the matrices
Therefore, due to the representation (47), the Eq. (52) yields
which, by virtue of (49), leads to
This proves (50). The inclusion (51) is obtained by similar considerations, based on the representation (48). \(\square \)
Let us denote
where
In other words, \(\varphi _u(t)\) is the minimal positive eigenvalue of the matrix \({\mathcal F}_u(t)\), while \(\varphi _v(t)\) is the maximal positive eigenvalue of the matrix \({\mathcal F}_v(t)\).
Proposition 6.3
The function \(\varphi _v(t)\) is continuous and decreases monotonically for \(t\in [t_0, t_f]\). Moreover,
Proof
Due to the representation (48) and the definition (58), the function \(\varphi _v(t)\) is continuous for all \(t\in [t_0, t_f]\), \(t\ne t_i\), \(i=1,\ldots ,K\). Moreover,
Thus, the function \(\varphi _v(t)\) is continuous.
Due to (48), \(m_t\in \mathrm{Ker}{\mathcal F}_v(t)\) iff the
and
This implies that, if \(\tau _1<\tau _2\), \(\tau _1, \tau _2\in [t_0, t_f[\), then
i.e., due to (59),
By virtue of the inclusion (66) and the definition (58), the function \(\varphi _v(t)\) decreases monotonically. \(\square \)
Proposition 6.4
The function \(\varphi _u(t)\) is continuous on the open intervals \(]t_i, t_{i+1}[\), \(i=1,\ldots , K\), and it is right-continuous for \(t=t_i\), \(i=1,\ldots , K\). Moreover,
Proof
The continuity of the function \(\varphi _u(t)\) on the intervals \(]t_i, t_{i+1}[\) is proved based on the representation (47), similarly to the proof of the continuity of the function \(\varphi _v(t)\) for \(t\in [t_0, t_f]\) based on the representation (48). Due to (47),
while, by virtue of (47) and (59),
\(\square \)
Now, let us proceed to the proof of the theorem.
Due to (15), Propositions 6.1–6.2 and by [10],
where the set \({\mathcal M}_t\) is given by (60),
If \(m_t\in \mathrm{Ker}{\mathcal F}_v(t)\), then, due to Propositions 6.1–6.2, \(g(m_t)\le 0\). Thus, the function \(g(m_t)\) can admit positive values only for \(m_t\in {\mathcal N}_t\), defined by (59). Therefore, in order to satisfy the condition (22), the inequality
should be valid for all \(t\in [t_0, t_f]\).
Due to (15), (57–58), (71) and (72),
Now, let us show that, for any fixed \(\beta >0\), it can be chosen a sufficiently small \(\alpha >0\), such that the following inequality holds:
The inequality (75), along with (74), guarantees (73).
The inequality (75) can be rewritten as
Remember that, due to Proposition 6.3, the function \(\varphi _v(t)\) is decreasing. Then, if \(\varphi _v(t_0)<\beta \), then the inequality (76) holds for all \(t\in [t_0, t_f]\) and for any \(\alpha >0\).
Let \(\varphi _v(t_0)\ge \beta \). Since the function \(\varphi _v(t)\) is continuous and due to (61), there exists a moment \(t^*=t^*(\beta )\in ]t_0, t_f[\) such that \(\varphi _v(t^*)=\beta \). Then, the condition (76) holds for all \(t\in ]t^*, t_f]\) and for any \(\alpha >0\). Thus, it is enough to prove that for a sufficiently small \(\alpha >0\), (76) holds for all \(t\in [t_0, t^*]\).
First, let us show that \(\alpha >0\) can be chosen in such a way that for all \(i=1,\ldots ,q(t^*)\),
Indeed, by virtue of (9–11), (15), (71) and (74),
Due to (70), the following values are defined correctly:
If
then
which, along with (78), guarantees (77). Moreover, the Eq. (71) and the inequalities (74) and (81) lead to
Similarly to Propositions 6.3–6.4, it can be proved that the function \(\lambda _{\alpha \beta }(t)\) is continuous on the open intervals \(]t_i, t_{i+1}[\), \(i=1,\ldots , K\), it is left-continuous for \(t=t_i\), \(i=1,\ldots , K\), and
Therefore, due to (77) and (82), there exist vicinities
such that, subject to (80),
for \(t\in T_1:=\bigcup \limits _{i=1}^{s(t^*)-1}d_i\). Let us denote
The set (86) consists of a finite number of closed intervals, where, due to Proposition 6.4, \(\varphi _u(t)>0\). Therefore, we can define
If
then the inequality (85) is valid for \(t\in T_2\). Now, let us define
Then, for all \(\alpha >0\), satisfying (23), the inequality (85) holds for all \(t\in [t_0, t^*]\), yielding (22). The saddle point (24–27) is derived based on Theorem 1 of [4]. This completes the proof of the theorem. \(\square \)
Appendix 2
1.1 Proof of Theorem 4.1
Let the trajectory \(x_{\alpha \beta }(t)\) be generated by the optimal strategy \(u_{\alpha \beta }^0(t,x)\), given by (25) for some \(\beta >0\) and \(\alpha <\alpha ^*(\beta )\), and by an arbitrary \(v(t)\in L_2^{m_v}[t_0,t_f]\), satisfying (4). Then, by the game value definition
leading, due to (2), to
Let us define
If \(\beta <\beta ^*\), then by (4),
Let \(\beta <\beta ^*(\zeta ,\nu )\) be fixed. Now, let us show that the game value \(J^0_{\alpha \beta }(t_0,x_0)\) also can be made arbitrary small by a proper choice of \(\alpha \). Due to Theorem 3.1 and Remark 3.1, there exists \(\tilde{\alpha }=\tilde{\alpha }(\beta )>0\) such that for \(\alpha \in ]0, \tilde{\alpha }[\), the matrix \(I_{Kn}-{\mathcal F}_{\alpha \beta }(t_0)\) is non-singular and
By virtue of (15) and by extracting \(\alpha \) from the inverse matrix in (94), the latter can be rewritten
Due to (29–30) and (50), the condition (44) means that
The latter, along with (95), yields
where \({\mathcal F}^\dag \) stands for a pseudo-inverse matrix of \({\mathcal F}\). The limiting expression (97) implies that for a fixed \(\beta >0\) and for an arbitrary \(\zeta >0\) there exists \(\bar{\alpha }=\bar{\alpha }(\beta ,\zeta )\in ]0, \tilde{\alpha }[\) such that for \(\alpha <\bar{\alpha }\),
By setting
and by using the inequalities (91), (93) and (98), we obtain that for \(\alpha <\alpha ^*(\zeta ,\nu )\), \(\beta <\beta ^*(\zeta ,\nu )\),
meaning that the strategy \(u_{\alpha \beta }^0(t,x)\) solves the Route Realization Problem. This completes the proof of the theorem. \(\square \)
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Turetsky, V. Robust Route Realization by Linear-Quadratic Tracking. J Optim Theory Appl 170, 977–992 (2016). https://doi.org/10.1007/s10957-016-0931-0
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DOI: https://doi.org/10.1007/s10957-016-0931-0