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Research Papers

# Existence of Solutions of Riccati Differential Equations

[+] Author and Article Information
Sinan Kilicaslan1 n2

Department of Mechanical Engineering, Faculty of Engineering, Gazi University, 06570 Maltepe, Ankara, Turkeyskilicaslan@gazi.edu.tr

Stephen P. Banks

Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield S1 3JD, United Kingdoms.banks@sheffield.ac.uk

1

Corresponding author.

2

Present address: Visiting Postdoctoral Scholar in the Department of Automatic Control and Systems Engineering of The University of Sheffield.

J. Dyn. Sys., Meas., Control 134(3), 031001 (Mar 27, 2012) (11 pages) doi:10.1115/1.4005496 History: Received March 12, 2010; Revised December 12, 2011; Published March 27, 2012; Online March 27, 2012

## Abstract

A necessary condition for the existence of the solution of the Riccati differential equation for both linear, time varying systems and nonlinear systems is introduced. First, a necessary condition for the existence of the solution of the Riccati differential equation for linear, time varying systems is proposed. Then, the sufficient conditions to satisfy the necessary condition are given. After that, the existence of the solution of the Riccati differential equation is generalized for nonlinear systems.

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Copyright © 2012 by American Society of Mechanical Engineers
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## Figures

Figure 1

Solution of p·=-rp2+q

Figure 2

State variables (first sv: solid line, second sv: dashed line)

Figure 3

State variables (first sv: solid line, second sv: dashed line)

Figure 4

State variables (first sv: solid line, second sv: dashed line)

Figure 5

State variables (first sv: solid line, second sv: dashed line)

Figure 6

Control input

Figure 7

Control input

Figure 8

Control input

Figure 9

Control input

Figure 10

Solution of ARE (P(1,1): solid line, P(1,2): dotted line, P(2,1): dashed-dotted line, P(2,2): dashed line)

Figure 11

Solution of ARE (P(1,1): solid line, P(1,2): dotted line, P(2,1): dashed-dotted line, P(2,2): dashed line)

Figure 12

Solution of ARE (P(1,1): solid line, P(1,2): dotted line, P(2,1): dashed-dotted line, P(2,2): dashed line)

Figure 13

Solution of ARE (P(1,1): solid line, P(1,2): dotted line, P(2,1): dashed-dotted line, P(2,2): dashed line)

Figure 14

Solution of RDE (P(1,1): solid line, P(1,2): dotted line, P(2,1): dashed-dotted line, P(2,2): dashed line)

Figure 15

Solution of RDE (P(1,1): solid line, P(1,2): dotted line, P(2,1): dashed-dotted line, P(2,2): dashed line)

Figure 16

Solution of RDE (P(1,1): solid line, P(1,2): dotted line, P(2,1): dashed-dotted line, P(2,2): dashed line)

Figure 17

Solution of RDE (P(1,1): solid line, P(1,2): dotted line, P(2,1): dashed-dotted line, P(2,2): dashed line)

Figure 18

Global convergence of first state variable (first iteration: dotted line, fourth iteration: dashed line, sixth iteration: dashed–dotted line, seventh iteration: solid line)

Figure 19

Global convergence of second state variable (first iteration: dotted line, fourth iteration: dashed line, sixth iteration: dashed–dotted line, seventh iteration: solid line)

Figure 20

Global convergence of control input (first iteration: dotted line, fourth iteration: dashed line, sixth iteration: dashed–dotted line, seventh iteration: solid line)

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