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Dejan Milutinovic - One of the best experts on this subject based on the ideXlab platform.
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DARS - Time Efficient Inspection of Ground Vehicles by a UAV Team Using a Markov Inequality Based Rule
Distributed Autonomous Robotic Systems, 2019Co-Authors: Alexey A Munishkin, Dejan Milutinovic, David W CasbeerAbstract:We present a control design for N unmanned aerial vehicles (UAVs) tasked with a time efficient inspection of M ground moving vehicles. The navigation and intent of each ground vehicle are unknown, therefore, the uncertainty of its navigation has to be anticipated in the navigation of each UAV. We use the minimum time stochastic optimal control to navigate each UAV towards the inspection of ground vehicles. Based on this control, we formulate assignments of ground vehicles to be inspected by UAVs as an optimization problem to inspect all ground vehicles in the minimum expected time. Accounting for ground vehicle uncertain trajectories, we update the optimal assignment by a Markov Inequality rule. The rule prevents the possibility of indefinite updating of assignments without finishing the inspection of all vehicles. On the other hand, it updates an assignment if it leads to a statistically significant improvement of the expected time of inspection. The presented approach is illustrated by a numerical example.
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time efficient inspection of ground vehicles by a uav team using a Markov Inequality based rule
Distributed Autonomous Robotic Systems, 2019Co-Authors: Alexey A Munishkin, Dejan Milutinovic, David W CasbeerAbstract:We present a control design for N unmanned aerial vehicles (UAVs) tasked with a time efficient inspection of M ground moving vehicles. The navigation and intent of each ground vehicle are unknown, therefore, the uncertainty of its navigation has to be anticipated in the navigation of each UAV. We use the minimum time stochastic optimal control to navigate each UAV towards the inspection of ground vehicles. Based on this control, we formulate assignments of ground vehicles to be inspected by UAVs as an optimization problem to inspect all ground vehicles in the minimum expected time. Accounting for ground vehicle uncertain trajectories, we update the optimal assignment by a Markov Inequality rule. The rule prevents the possibility of indefinite updating of assignments without finishing the inspection of all vehicles. On the other hand, it updates an assignment if it leads to a statistically significant improvement of the expected time of inspection. The presented approach is illustrated by a numerical example.
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Markov Inequality rule for switching among time optimal controllers in a multiple vehicle intercept problem
Automatica, 2018Co-Authors: Dejan Milutinovic, David W Casbeer, Meir PachterAbstract:Abstract In this paper, a Markov Inequality based switching rule is proposed to switch among numerically computed, time optimal controllers in a multiple vehicle intercept problem. Each controller is optimal for the intercept of a single vehicle, i.e., for the segment of the complete time varying multiple vehicle target set. The switching rule guarantees that after every switch the time to the target set is shorter with a certain predefined probability. Furthermore, the rule guarantees that the target set is reached after a finite number of switches and the rule scales well with the number of vehicles, i.e., the segments covering the target set. The problem and results are illustrated by a numerical example.
David W Casbeer - One of the best experts on this subject based on the ideXlab platform.
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DARS - Time Efficient Inspection of Ground Vehicles by a UAV Team Using a Markov Inequality Based Rule
Distributed Autonomous Robotic Systems, 2019Co-Authors: Alexey A Munishkin, Dejan Milutinovic, David W CasbeerAbstract:We present a control design for N unmanned aerial vehicles (UAVs) tasked with a time efficient inspection of M ground moving vehicles. The navigation and intent of each ground vehicle are unknown, therefore, the uncertainty of its navigation has to be anticipated in the navigation of each UAV. We use the minimum time stochastic optimal control to navigate each UAV towards the inspection of ground vehicles. Based on this control, we formulate assignments of ground vehicles to be inspected by UAVs as an optimization problem to inspect all ground vehicles in the minimum expected time. Accounting for ground vehicle uncertain trajectories, we update the optimal assignment by a Markov Inequality rule. The rule prevents the possibility of indefinite updating of assignments without finishing the inspection of all vehicles. On the other hand, it updates an assignment if it leads to a statistically significant improvement of the expected time of inspection. The presented approach is illustrated by a numerical example.
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time efficient inspection of ground vehicles by a uav team using a Markov Inequality based rule
Distributed Autonomous Robotic Systems, 2019Co-Authors: Alexey A Munishkin, Dejan Milutinovic, David W CasbeerAbstract:We present a control design for N unmanned aerial vehicles (UAVs) tasked with a time efficient inspection of M ground moving vehicles. The navigation and intent of each ground vehicle are unknown, therefore, the uncertainty of its navigation has to be anticipated in the navigation of each UAV. We use the minimum time stochastic optimal control to navigate each UAV towards the inspection of ground vehicles. Based on this control, we formulate assignments of ground vehicles to be inspected by UAVs as an optimization problem to inspect all ground vehicles in the minimum expected time. Accounting for ground vehicle uncertain trajectories, we update the optimal assignment by a Markov Inequality rule. The rule prevents the possibility of indefinite updating of assignments without finishing the inspection of all vehicles. On the other hand, it updates an assignment if it leads to a statistically significant improvement of the expected time of inspection. The presented approach is illustrated by a numerical example.
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Markov Inequality rule for switching among time optimal controllers in a multiple vehicle intercept problem
Automatica, 2018Co-Authors: Dejan Milutinovic, David W Casbeer, Meir PachterAbstract:Abstract In this paper, a Markov Inequality based switching rule is proposed to switch among numerically computed, time optimal controllers in a multiple vehicle intercept problem. Each controller is optimal for the intercept of a single vehicle, i.e., for the segment of the complete time varying multiple vehicle target set. The switching rule guarantees that after every switch the time to the target set is shorter with a certain predefined probability. Furthermore, the rule guarantees that the target set is reached after a finite number of switches and the rule scales well with the number of vehicles, i.e., the segments covering the target set. The problem and results are illustrated by a numerical example.
Szilárd Gy. Révész - One of the best experts on this subject based on the ideXlab platform.
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On a paper of Erod and Turan-Markov inequalities for non-flat convex domains
arXiv: Classical Analysis and ODEs, 2007Co-Authors: Szilárd Gy. RévészAbstract:For a convex domain K in the complex plane C, the well-known general Markov Inequality asserting that a polynomial p of degree n ||p'|| < c(K) n^2 ||p|| holds. On the other hand for polynomials in general, ||p'|| can be arbitrarily small as compared to ||p||. The situation changes when we assume that the polynomials have all their zeroes in the convex body K. This problem of lower bound for Markov factors was first investigated by Tur\'an in 1939. Tur\'an showed ||p'|| \ge n/2 ||p|| for the unit disk D and ||p'|| > c \sqrt{n} ||p|| for the unit interval I:=[-1,1]. Soon after that, J. Er\H od published a long article, discussing various extensions of the results and methods of Tur\'an. For decades, Er\H od's paper was quoted only for the explicit calculation of the exact constant of the interval case. However, in recent years Levenberg and Poletsky, Erd\'elyi and also the author investigated Tur\'an's problem for various sets - basically, convex domains. In this context the much richer content of Er\H od's work is to be realized again. Thus, the aim of the paper is twofold. On the one hand we give an account of the half-forgotten, old Hungarian article of Er\H od, also commemorating its author. On the other hand we report on recent developments with particular emphasis on development of one of the key observations of Er\H od, namely, the role of the curvature of the boundary curve in the estimation of the lower bound of Markov factors.
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Turán type reverse Markov inequalities for compact convex sets
Journal of Approximation Theory, 2006Co-Authors: Szilárd Gy. RévészAbstract:For a compact convex set K ⊂ C the well-known general Markov Inequality holds asserting that a polynomial p of degree n must have ||p'|| ≤ c(K)n2 ||p||. On the other hand for polynomials in general, ||p'|| can be arbitrarily small as compared to ||p||.The situation changes when we assume that the polynomials in question have all their zeroes in the convex set K. This was first investigated by Turan, who showed the lower bounds ||p'|| ≥ (n/2) ||p|| for the unit disk D and ||p'|| ≥ c√n ||p|| for the unit interval I:= [-1, 1]. Although partial results provided general lower estimates of order √n, as well as certain classes of domains with lower bounds of order n, it was not clear what order of magnitude the general convex domains may admit here. Here we show that for all bounded and convex domains K with nonempty interior and polynomials p with all their zeroes lying in K ||p'|| ≥ c(K)n ||p|| holds true, while ||p'|| ≤ C(K)n ||p|| occurs for any K. Actually, we determine c(K) and C(K) within a factor of absolute numerical constant.
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Right order Turan-type converse Markov inequalities for convex domains on the plane
arXiv: Classical Analysis and ODEs, 2005Co-Authors: Szilárd Gy. RévészAbstract:For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov Inequality holds asserting that a polynomial $p$ of degree $n$ must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in general, $||p'||$ can be arbitrarily small as compared to $||p||$. The situation changes when we assume that the polynomials in question have all their zeroes in the convex body $K$. This was first investigated by Tur\'an, who showed the lower bounds $||p'|| \ge (n/2) ||p||$ for the unit disk $D$ and $||p'|| > c \sqrt{n} ||p||$ for the unit interval $I:=[-1,1]$. Although partial results provided general lower estimates of lower order, as well as certain classes of domains with lower bounds of order $n$, it was not clear what order of magnitude the general convex domains may admit here. Here we show that for all compact and convex domains $K$ with nonempty interior and polynomials $p$ with all their zeroes in $K$ $||p'|| > c(K) n ||p||$ holds true, while $||p'|| < C(K) n ||p||$ occurs for any $K$. Actually, we determine $c(K)$ and $C(K)$ within a factor of absolute numerical constant.
Raimondo Eggink - One of the best experts on this subject based on the ideXlab platform.
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Equivalence of the global and local Markov inequalities in the complex plane
Advances in Mathematics, 2019Co-Authors: Leokadia Bialas-ciez, Raimondo EgginkAbstract:Abstract We prove that a compact subset of the complex plane satisfies the Global Markov Inequality if and only if it admits an extension property and a Kolmogorov type Inequality in Jackson norms, generalizing a result established by Bos and Milman in the real case. We also show that the Global Markov Inequality is equivalent to the Local Markov Property for all compact subsets of the complex plane admitting the Jackson Property. The latter is a generalization of Jackson's approximation Inequality, closely connected with the boundary behavior of the Green's function. Finally, we construct an example showing that there is no equivalence in general.
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Global and Local Markov Inequalities in the Complex Plane
Dolomites Research Notes on Approximation, 2014Co-Authors: Leokadia Bialas-ciez, Raimondo EgginkAbstract:We present the current state of the art concerning the global and local Markov inequalities in the complex plane. This paper is based on a talk given during the Workshop on Multivariate Approximation in honor of Prof. Len Bos 60th birthday, and rests on articles [4], [5] and [6]. Our research is inspired by papers by Bos and Milman where global and local Markov inequalities are compared in the real case (see [7], [8]). We are interested in obtaining analogous results in the complex plane in view of further investigation of properties of the Green’s function connected with Markov sets. We first recall two crucial results obtained by Bos and Milman (see [9]). Theorem A. Suppose that E R n is compact. Then a local Markov Inequality with exponent r 1 is equivalent to a Geometric Inequality with the same exponent r 1 and implies a Sobolev Inequality (with Whitney norms), also with the same exponent r. A Sobolev Inequality in Whitney norm implies a Sobolev Inequality in the quotient norm. Conversely, if E admits a Sobolev Inequality in the quotient norm with exponent r 1, then E admits a local Markov Inequality with any exponent > r. Moreover, in the regular case, r = 1, we may take = r = 1. Theorem B. Suppose that E R n is compact and C 1 -determining. Then the following are equivalent: 1. E admits a Sobolev type Inequality in quotient norm,
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Equivalence of the Local Markov Inequality and a Kolmogorov Type Inequality in the Complex Plane
Potential Analysis, 2012Co-Authors: Leokadia Bialas-ciez, Raimondo EgginkAbstract:We prove that a compact subset of the complex plane satisfies a local Markov Inequality if and only if it satisfies a Kolmogorov type Inequality. This result generalizes a theorem established by Bos and Milman in the real case. We also show that every set satisfying the local Markov Inequality is a sum of Cantor type sets which are regular in the sense of the potential theory.
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L-Regularity of Markov Sets and of m-Perfect Sets in the Complex Plane
Constructive Approximation, 2008Co-Authors: Leokadia Bialas-ciez, Raimondo EgginkAbstract:Let E be a compact subset of C . We prove that if E satisfies the following local Markov property: for each polynomial P, $|P'(z_0)|\le \frac{M({\rm deg}\, P)^s}{r^m}\max \{ |P(z)| : z\in E \ {\rm and} \ |z-z_0|\le r\},$ where M, m, s are positive constants independent of P, $r\in (0,1]$ and $z_0\in E$ ; then E is L-regular, i.e. regular in the sense of the potential theory. In particular, if $E\subset \mbox{\bf R}$ satisfies the global Markov Inequality, then E is L-regular. We also prove that if $E\subset \mbox{\bf C}$ is an m-perfect set (there exists c > 0 such that, for all $z_0\in E$ and $r\in (0,1]$, $\left.\left\{ z\in E : \frac{r^m}{c}\le |z-z_0|\le r\right\}\ne \emptyset \right)$ and $m\in [1,2)$ , then E is L-regular. Examples given by Siciak [20] show that the assumption that m < 2 cannot be omitted.
Meir Pachter - One of the best experts on this subject based on the ideXlab platform.
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Markov Inequality rule for switching among time optimal controllers in a multiple vehicle intercept problem
Automatica, 2018Co-Authors: Dejan Milutinovic, David W Casbeer, Meir PachterAbstract:Abstract In this paper, a Markov Inequality based switching rule is proposed to switch among numerically computed, time optimal controllers in a multiple vehicle intercept problem. Each controller is optimal for the intercept of a single vehicle, i.e., for the segment of the complete time varying multiple vehicle target set. The switching rule guarantees that after every switch the time to the target set is shorter with a certain predefined probability. Furthermore, the rule guarantees that the target set is reached after a finite number of switches and the rule scales well with the number of vehicles, i.e., the segments covering the target set. The problem and results are illustrated by a numerical example.