Stability conditions in a water hammer model involving two delays

The linear model of a hydroelectric power plant under water hammer is considered. This model contains the hyperbolic Partial Differential Equations (hPDEs) of wave propagation along the distributed parameter tunnel and penstock, also a surge tank with throttling. The water hammer occurs following hydraulic turbine shut down. A system of Functional Differential Equations of neutral type (nFDEs) is associated to the basic Boundary Value Problem (BVP) and asymptotic stability is obtained using a quadratic Lyapunov functional suggested by the energy identity.

[1]  Daniela Danciu,et al.  Control of a Time Delay System Arising From Linearized Conservation Laws , 2019, IEEE Access.

[2]  Guenter Leugering,et al.  On the Modelling and Stabilization of Flows in Networks of Open Canals , 2002, SIAM J. Control. Optim..

[3]  Martin Gugat,et al.  Optimal Boundary Control and Boundary Stabilization of Hyperbolic Systems , 2015, Springer Briefs in Electrical and Computer Engineering.

[4]  Charles Jaeger,et al.  Fluid transients in hydro-electric engineering practice , 1977 .

[5]  Kamel Baddari.  Equations de la physique mathématique appliquées , 2009 .

[6]  M. Krstić.  Boundary Control of PDEs: A Course on Backstepping Designs , 2008 .

[7]  Jonathan de Halleux,et al.  Boundary feedback control in networks of open channels , 2003, Autom..

[8]  G. Bastin,et al.  Stability and Boundary Stabilization of 1-D Hyperbolic Systems , 2016 .

[9]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[10]  Georges Bastin,et al.  A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws , 2007, IEEE Transactions on Automatic Control.

[11]  R. Datko.  A procedure for determination of the exponential stability of certain differential-difference equations , 1978 .

[12]  M. Herty,et al.  EXISTENCE OF CLASSICAL SOLUTIONS AND FEEDBACK STABILIZATION FOR THE FLOW IN GAS NETWORKS , 2011 .

[13]  H. Antosiewicz,et al.  Differential Equations: Stability, Oscillations, Time Lags , 1967 .

[14]  V. Răsvan.  Control of Conservation Laws – An Application , 2018, Tatra Mountains Mathematical Publications.

[15]  Günter Leugering,et al.  Classical solutions and feedback stabilization for the gas flow in a sequence of pipes , 2010, Networks Heterog. Media.

[16]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.