Index. Chromatography, 3 Condition of C 0 compatibility, 8, 9 Condition of C 1 compatibility, 8, 9, 33, 36, 38, 39, 41, 43, 50, 53, 56, 58, 59, 62

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1 References 1. Bressan, A., Čanić, S., Garavello, M., Herty, M., Piccoli, B.: Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1, (2014) 2. Coron, J.-M., Wang, Z.: Controllability for a scalar conservation law with nonlocal velocity. J. Differ. Equ. 252, (2012) 3. D Apice, C., Göttlich, S., Herty, M., Piccoli, B.: Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, Society for Industrial and Applied Mathematics (SIAM). Philadelphia, PA (2010) 4. de Saint-Venant, B.: Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et l introduction des marées dans leur lit, C. R. Acad. Sci. 73, , (1871) 5. Garavello, M., Piccoli, B.: Traffic flow on networks. AIMS Ser. Appl. Math. 1 (2006). American Institute of Mathematical Sciences (AIMS), Springfield, MO 6. GU, Q., Li, T.: Exact boundary controllability for quasilinear wave equations in a planar treelike network of strings. Ann. de L Institut Henri Poincaré Anal. Non Linéaire 26, (2009) 7. GU, Q., Li, T.: Exact boundary controllability for quasilinear hyperbolic systems on a tree-like network and its applications. SIAM J. Control Optim. 49, (2011) 8. Gu, Q., Li, T.: Exact boundary controllability of nodal profile for quasilinear hyperbolic systems in a tree-like network. Math. Methods Appl. Sci. 34, (2011) 9. Gu, Q., Li, T.: Exact boundary controllability of nodal profile for unsteady flows on a tree-like network of open canals. J. de Mathématiques Pures et Appliquées 99, (2013) 10. Gugat, M., Herty, M., Schleper, V.: Flow control in gas networks: exact controllability to a given demand. Math. Methods Appl. Sci. 34, (2011) 11. Li, T.: Controllability and observability: From ODEs to quasilinear hyperbolic systems. In: Jeltsch, R., Wanner, G. (eds.) Sixth International Congress on Industrial and Applied Mathematics (ICIAM 07), Zürich, Switzerland, July Invited Lectures, European Mathematical Society, 2009, Li, T.: Controllability and observability for quasilinear hyperbolic systems. AIMS Series on Applied Mathematics, vol. 3. American Institute of Mathematical Sciences & Higher Education Press (2010) 13. Li, T.: Exact boundary controllability of nodal profile for quasilinear hyperbolic systems. Math. Methods Appl. Sci. 33, (2010) 14. Li, T.: A constructive method to controllability and observability for quasilinear hyperbolic systems. Methods Appl. Anal. 18, (2011) 15. Li, T., Jin, Y.: Semi-global C 1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems. Chin. Ann. Math. Ser. B 22, (2001) The Author(s) 2016 T. Li et al., Exact Boundary Controllability of Nodal Profile for Quasilinear Hyperbolic Systems, SpringerBriefs in Mathematics, DOI /

2 104 References 16. Li, T., Rao, B.: Local exact boundary controllability for a class of quasilinear hyperbolic systems. Chin. Ann. Math. Ser. B 23, (2002) 17. Li, T., Rao, B.: Exact boundary controllability for quasilinear hyperbolic systems. SIAM J. Control Optim. 41, (2003) 18. Li, T., Rao, B.: Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems. Chin. Ann. Math. Ser. B 31, (2010) 19. Li, T., Yu, L.: Exact boundary controllability for 1-D quasilinear wave equations. SIAM J. Control Optim. 45, (2006) 20. Li, T., Yu, W.: Boundary Value Problems for Quasilinear Hyperbolic Systems. Duke University Mathematics Series V (1985) 21. Wang, K.: Exact boundary controllability of nodal profile for 1-D quasilinear wave equations. Frontiers Math. China 6, (2011) 22. Wang, K., Gu, Q.: Exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree-like network of strings. Math. Methods Appl. Sci. 37, (2014) 23. Zhuang, K.: Exact controllability of nodal profile for 1-D first order quasi-linear hyperbolic systems with zero eigenvalues, Preprint 24. Zhuang, K.: Exact controllability of nodal profile with internal controls for 1-D first order quasilinear hyperbolic systems, Preprint

3 Index A Artificial boundary condition, 36, 38, 56 59, 61, 63, 78 81, 87, 88, 90, 93, 95 Artificial flux boundary condition, 54 Artificial multiple node, 72 B Backward mixed initial-boundary value problem, 10 Blood flow, 3 Boundary condition, 7 11, 13, 16 20, 22, 25, 28, 32, 33, 36 39, 41, 43, 45, 47, 49 52, 54, 57, 59 61, 63, 67 69, 76 82, 84 91, 93 95, 97, 99, 100 Boundary control, 33 35, 37, 41 43, 46, 47, 50, 52 58, 61, 62, 64, 66, 67, 69 72, 76 78, 82, 83, 85, 86, 91, 93, 95, 96, , 102 Boundary control function, 68, 80, 82, 90, 100, 101 Boundary flux function, 50, 53 Boundary function, 76, 77, 85, 91, 98, 101 Boundary node, 31, 33, 34, Chromatography, 3 Condition of C 0 compatibility, 8, 9 Condition of C 1 compatibility, 8, 9, 33, 36, 38, 39, 41, 43, 50, 53, 56, 58, 59, 62 Condition of C 1 or piecewise C 1 compatibility, 25, 27 Condition of C 2 compatibility, 16, 17, 76, 77, 79 81, 85, 87, 88, 90, 91, 93, 95 Condition of C 2 or piecewise C 2 compatibility, 29, 92, 98 Condition of piecewise C 1 compatibility, 55, 66 Condition of piecewise C 2 compatibility, 86, 96 Constraint, 41, 42, 45, 54, 55, 57, 61, 62, 64, 69, 85, 91, 96 Constructive method, 78, 86, 93 Constructive method with modular structure, 31, 35, 75 Continuity of displacement, 29, 85 Control, 41, 47 Control function, 85 Controllability time, 66, 69 C C 1 norm, 27 C 1 solution, 7, 11, 33 39, 41 43, 46, 50 52, 54, 57, 59 61, 63, 64, 68 C 2 solution, 20, 21, 76 82, 88 90, 94, 95, 99 Canal, 42 Characteristic, 7, 18 Characteristic form, 6 Characteristic form of hyperbolic system, 3 The Author(s) 2016 T. Li et al., Exact Boundary Controllability of Nodal Profile for Quasilinear Hyperbolic Systems, SpringerBriefs in Mathematics, DOI / D Diagonal variable, 4, 8, 19, 25, 26, 47 Dirichlet type, 16, 28, 76, 84 Dissipative type, 16, 28, 76, 84 Downmost characteristic, 12, 20 E Eigenvalue, 49 Energy-type interface condition, 49, 50, 53 55, 60, 63, 68,

4 106 Index Equilibrium, 32, 40 Exact boundary controllability, 31, 83 Exact boundary controllability of nodal profile, 11, 16, 19, 31, 33 35, 45 48, 53, 55, 58, 62, 64, 66, 70 73, 76, 77, 82, 83, 91, Exact boundary controllability of nodal profile on a boundary node, 33, 76 Exact boundary controllability of nodal profile on an internal node, 34, 77 Exact boundary controllability of nodal profile on the internal node, 35, 77 Exact controllability of nodal profile, 46, 73 F Final condition, 10 First order quasilinear hyperbolic system, 7, 23, 24, 29 First order quasilinear system, 17 Flux boundary condition, 43, 50, 52, 58, 69 Flux function, 57 Forward mixed initial-boundary value problem, 7, 8, 10, 16, 25, 36, 38, 60, 63, 78 80, 87, 89, 93, 94, 99 Forward mixed problem, 80, 87, 89, 93, 94 Forward problem, 36, 38 G General 1-D first order quasilinear hyperbolic system, 47 General 1-D quasilinear hyperbolic equations (systems) of second order, 82, 102 General 1-D quasilinear hyperbolic system, 25, 31 General nonlinear boundary condition, 31, 47 General nonlinear interface condition, 47 Global exact boundary controllability of nodal profile, 33 H Hyperbolic system, 1 Hyperbolicity, 26, 32 57, 59 61, 63 65, 67, 68, 76, 78 82, 84, 87 90, 93 95, 97, 99, 100 Initial data, 11, 20, 33, 34, 40, 42, 43, 46, 50, 53, 55, 58, 62, 63, 76 78, 85, 86, 91, 92, 96 Interface condition, 25, 28, 47, 56, 57, 61, 63, 67 69, 71 73, 84, 87, 89 94, Interface conditions at the multiple node, 64 Internal control, 46, 82 Internal node, 31, 34, 42, 46, 71, 72, 75, 77, 78, 101, 102 Irrigation channel, 3 J Joint node, 84, 85, 98, 99 K Kronecker symbol, 2 L Left eigenvector, 1, 2, 18, 26, 32 Leftmost characteristic, 11, 20 Leftward mixed initial-boundary value problem, 11, 36, 38, 59, 79, 81, 88 Leftward mixed problem, 59, 79, 81, 88 Leftward one-sided mixed initial-boundary value problem, 82 Leftward problem, 36, 39 Local exact boundary controllability, 31, 75 Local exact boundary controllability of nodal profile, 7, 33, 75 Local exact boundary controllability of nodal profile on an internal node, 34 M Maximum determinate domain, 11 13, 20, 21, 37, 39, 80, 82, 89, 90, 95 Mixed initial-boundary value problem, 7, 8, 27, 29, 33 35, 41 43, 46, 50, 52, 54, 56, 58, 62, 76 78, 85, 86, 91, 93, 96 Mixed problem, 100 Multiple node, 23 26, 28, 48 50, 52, 53, 55, 57, 62, 64 73, 83, 84, 87, 89, 91 93, I Incoming characteristic, 7 10, 25 Initial condition, 7, 8, 11, 13, 15 18, 22, 25, 27, 28, 32, 36 41, 43, 49 52, 54, 56, N Network, 23 Network of open canals, 67 Neumann type, 16, 28, 76, 84

5 Index 107 Nodal profile, 41 43, 45 47, 50, 52 58, 61, 62, 64, 66 72, 75, 83, 85, 91, 96, Nodal profile control, 31 Node, 29 Nonlinear boundary condition, 24, 26, 40 Nonlinear interface condition, 24, 26 O One-dimensional (1-D) elastic wave, 3 One-dimensional (1-D) first order quasilinear hyperbolic system, 1, 23, 32 One-dimensional (1-D) first order quasilinear system, 1 One-dimensional (1-D) gas dynamics without viscosity, 3 One-dimensional (1-D) quasilinear hyperbolic system, 3, 31 One-dimensional (1-D) quasilinear wave equation, 15, 23, 27, 75, 83, 84, 91 One-sided exact boundary controllability, 34 One-sided forward mixed initial-boundary value problem, 11, 20 One-sided forward mixed problem, 11 One-sided leftward mixed initial-boundary value problem, 13, 22, 37, 39 One-sided mixed initial-boundary value problem, 11, 19, 80, 88 One-sided mixed problem, 37, 39, 80, 82, 90, 95 One-sided rightward mixed initial-boundary value problem, 11, 13, 20, 21, 39 Open canal, 48 Optimal controllability time, 35, 78 Outgoing characteristic, 8 10 P Piecewise C 1 norm, 25 Piecewise C 1 solution, 25, 27, 50, 52, 54, 56 58, 61 63, 67, 68 Piecewise C 2 solution, 29, 85 87, 90, 91, 93, 95, 96, 99, 100 Planar star-like network of strings, 83 Planar tree-like network, 97, 101 Planar tree-like network of strings, 83, 97, 98, 102 Q Quasilinear hyperbolic system of diagonal form, 40 Quasilinear wave equation, 16, 19, 97, 101, 102 R Real eigenvalue, 1, 2, 18, 26, 32 Reducible quasilinear hyperbolic system, 4 Riemann invariant, 5, 6, 23, 44, 51 Right eigenvector, 1, 2 Rightward mixed initial-boundary value problem, 11, 38, 56, 61, 63, 81, 90, 95 Rightward mixed problem, 57, 61, 81, 90, 95 Rightward one-sided mixed initial-boundary value problem, 82 Rightward problem, 39 S Saint-Venant system, 6, 47, 48, 53, 55, 58, 62, 64, 65, Saint-Venant system for unsteady flows, 5, 23, 42 Semi-global C 1 solution, 7, 8, 10 Semi-global C 2 solution, 16 Semi-global classical solution, 23 Semi-global piecewise classical solution, 23 Simple node, 23 26, 28, 47, 49, 50, 52, 53, 55, 57, 58, 61, 62, 64 71, 83 88, 91, 93, Single interval, 23 Single node, 52 Single open canal, 5, 42, 45, 71 Single spatial interval, 47 Single string, 83, 99, 102 Small C 1 solution, 13 Small C 2 solution, 21 Small piecewise C 1 solution, 27 Small piecewise C 2 norm, 29 Star-like network, 48, 52, 66, 67, 71, 72, 83, 84, 101 Star-like network of open canals, 48, 53, 55, 58, 62 Star-like network of strings, 83, 87, 91, 93 Strictly hyperbolic system, 2, 18 String, 23 25, 27 String-like network, 100, 101 Subcritical case, 5 Subcritical equilibrium state, 5, 43, 49 Subcritical region, 72 Subnetwork, 67, 68, 71, 72, 99, 100, 102 Supply chains, 3 System of diagonal form, 44

6 108 Index T Third type, 16, 28, 76, 84 Total flux, 53, 55, 64 Total flux function, 71, 72 Total flux interface condition, 49, 50, 60, 63, 64, 66, 68, 69, 71 Total stress, 85 Total stress at the multiple node, 29, 98 Total stress function, 85, 96, Traffic flow, 3 Transfer of boundary controls, 70 Tree-like network, 23, 27, 31, 47, 64, 66, 67, 71 73, , 102 Tree-like network of open canals, 47, 64, 66, Tree-like network of strings, 83, 98 Two-sided exact boundary controllability, 35 U Uniqueness of C 1 solution, 11 Uniqueness of C 1 solution to the one-sided mixed problem, 37, 39 Uniqueness of C 2 solution, 19, 80, 82 Unsteady flow, 47 Unsteady flow on a star-like network of open canals, 48, 64 W Wave equation, 99 Well-posedness, 7, 32 Z Zero eigenvalue, 7, 9, 10, 73, 82, 102