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Typically, active control systems either have a priori complete information about the boundary-value problem and damped waves before switching on, or get it during the measurement process or accumulate and update information online (identification process in adaptive systems). In this case, the boundary problem is completely imprinted in the information arrays of the control system. However, very often complete information about a boundary-value problem is not available in principle or this info is changing in time faster than the process of its accumulation. The article considers examples of boundary control algorithms based almost without any information. The algorithms presented in the article cannot be obtained within the framework of the harmonic representation of the problem by complex amplitudes. And these algorithms carry out fast control in microstructured boundary problems. It is shown that in some cases it is possible to find simple solutions if we remove restrictions: 1) on the spatio-temporal resolution of controlling elements of a boundary-value problem; 2) on the high-frequency radiation of the controlled boundary.

With constant (in time) parameters (or without frequency conversion) devices for wideband non-resonant sound suppression (or other types of waves) should have large wave sizes (thickness l D , see

We mean breaking as a very quick and microscopic hop (jump) from one boundary value problem to another. Below we want to obtain: (a) effective suppression of long wave reflections (

The goal of the algorithm is to suppress reflections from boundary x b . We consider a semi-infinite ( x b ≤ x < ∞ ) elastic rod, with longitudinal impedance Z,

sound speed c of waves and the field U ( x , t ) of longitudinal displacement of particles (

The boundary-value problem can be represented as the sum of two partial linear problems: (a1) reflection of the incident wave (IW) U W ≠ 0 from the free (at F b = 0 ) the end of the rod; (a2) wave generation by force F b ≠ 0 in the absence of an incident wave (at U W = 0 ). The “breaker” jumps ((a1) ↔ (a2)) in accordance with the control algorithm: measurement in (a1), action in (a2). We assume that the force F b has a compact support: | F b | > 0 at t ∈ [ 0 , τ F ] , F b = 0 at t ∉ [ 0 , τ F ] . For a clear distinction between the causes and consequences in the work of AHRB, it is extremely important that after the termination of the force F b (at t > τ F ), the displacement ∫ 0 τ F Z − 1 F b ( ξ ) d ξ of the boundary x b caused by this force is saved indefinitely long [

φ [ ξ ] = ∫ t n − 1 + ( τ f ) n t n − 1 + ( τ f ) n + ξ Z − 1 F b ( η ) d η for the first time (at the value ξ = ( τ r ) n ) beginning

from the moment t = t n − 1 + ( τ f ) n . It is easy to see from the

U ¯ W = ( 1 / τ W c ) ∫ 0 τ W c | U W ( ξ ) | d ξ . The peak P ^ h f and average P ¯ h f (on the cycle ~T) power of the high-frequency radiation, generated by the impact force F b , and time averaged power flux P ¯ W in the low frequency IW are satisfying to the following relations P ^ h f / P ¯ W ≈ ( τ f / τ r ) 2 > > 1 and P ¯ h f / P ¯ W ≈ τ f / τ r > > 1 . About impedance: Z is unknown and can slowly change in time Z ( t ) . About impact force: F b ( t ) can be of arbitrary pulse shape, but with constant sign and at any moment of impact satisfies the condition F ¯ b > > Ψ ¯ W , where

F ¯ b = ( 1 / τ W ) ∫ 0 τ W F b ( t ) d t , Ψ ¯ W = ( ε S / τ W ) − 1 ∫ 0 τ W c | ( U W ( x + с t ) ) x / | d x . About linearity: the returning path scale is h = ( τ f / τ W ) U ¯ W , velocity scale is h / τ r , so the condition h / τ r < < c ensures linearity.

The goal of the algorithm AMIP is to maximize the instantaneous power absorbed by the boundary x b . Consider above rod problem: some electric drive (as above “breaker”) can ensure any constant velocity V b of edge x b independently of any incident wave (IW). Wave problem can be represented as the sum of two linear problems: (a1) reflection of IW from a fixed ( V b = d U b / d t = 0 ) boundary x b ; (a2) the radiation of waves by a boundary x b at a given velocity V b in the absence of IW. The breaker jumps: (a1) ↔ (a2). Velocity V b takes discrete levels V b ( t n ) = V n at discrete time intervals ( n − 1 ) T < t < n T ( n = 1 , 2 , ... ), where T is the period of velocity switching (and measuring between switching). Steps V n are multiples of the tuning step V ¯ , V n / V ¯ = 0 , ± 1 , ± 2 , ± 3 , ... (integer).

Absorbed power (work of IW with a border x b ) is square function W ( t ) = V b ( t ) [ 2 E W ( x b + c t ) − Z V b ( t ) ] of V ( t ) with the unique maximum at V b ( t ) = V W ( x b + c t ) = E W ( x b + c t ) / Z , where Z = Re Z , V W ( x b + c t ) and E W ( x b + c t ) are longitudinal particle velocity and stress in IW in infinite rod. AMIP is expressed by the iterative (recurrent) relation V n = V n − 1 + V ¯ sgn ( W ¯ ) (for n ≥ 2 with initial condition V 1 = 0 , V 1 = V ¯ ), where: W ¯ = W n − 1 − W n − 2 , W n − 1 = F n − 1 V n − 1 , W n − 2 = F n − 2 V n − 2 ; F n − 1 , F n − 2 are measured values of the force applied to boundary x b by the medium of rod from x ≥ x b at the moments t n − 1 − a , t n − 2 − a correspondingly ( 0 < a < < T ); sgn ( ξ ) = + 1 at ξ > 0 , sgn ( ξ ) = − 1 at ξ < 0 . If at the previous step the velocity increase causes the decrease ( W ¯ < 0 ) of the absorbed power, at the next step the velocity increase will change its sign and will not change it in the opposite case. In above one dimensional statement of the problem absorption maximum corresponds to the minimum of reflection and radiation too. AMIP does not need to know either rod impedance Z and IW. AMIP effectively traces IW if the following conditions are satisfied: V ¯ < < max | ∂ U W / ∂ t | , V ¯ / T > > max | ∂ 2 U W / ( ∂ t ) 2 | or τ W V ¯ < < U ¯ W < < ( τ W ) 2 T − 1 V ¯ , AMIP resembles the algorithm of random search, considered in [

The goal of the algorithm is to suppress reflections from boundary x b in the above one-dimensional problem and in layer (or distance from x b ) of small thickness at minimum info on IW and boundary problem. The ABCM is based on two main states of a controlled boundary x b (

wave into reflected high-frequency waves at frequencies f = 1 / T , 2 / T , 3 / T , ... that dissipatively attenuate in exp [ − l / l D ( f ) ] times at a distance l from the boundary x b , where l D = l D ( f ) is the frequency dependent dissipative attenuation length or equivalent damping device size (see

Thus, having fulfilled the condition c T < < l D ( 1 / T ) < < τ W c < < l D ( 1 / τ W ) , it is possible to ensure the smallness of the attenuation length and smallness of effect of dissipation on the above boundary condition.

The goal of ASBG is to block the sound propagation in a gas stream with average velocity υ < < c . This problem arises in the design of automobile silencers (

Consider the 1D case (

Next are a lot (Figures 5(a2)-(a4)) of inverters-breakers in the system cross section, now a pair of mutually complementary ( B ¯ ( t ) B ¯ ¯ ( t ) = 0 ) inverter breakers B ¯ ( t ) , B ¯ ¯ ( t ) (see

The goal is to reduce the ringingness of a tank (as a resonator for surface water waves) without increasing the viscosity of waveguiding media (water). In the traditional case of time constant parameters (Section 1,

On the left edge of the tank (

The vertical shift H ( t ) of the free surface of the water near the wall L is measured by a sensor in the form of a float. The friction breaker B ( t ) is switching periodically (with an interval T / 2 = 1 / 2 f M , where f M the frequency of binary modulation) between two states: (a) “stopped” state ( B ( t ) = 1 , the breaker strongly pressing to the upper edge L and fixes the angle of deviation

of the wall L ); (b) “free” state ( B ( t ) = 0 the breaker does not touch the upper edge L ) states of the wall L . Hydrostatic pressure on the wall L on the right is compensated by a soft elastic spring; the softness of spring is such that frequency f 0 of free (at B ( t ) = 0 ) oscillations of the wall L is less than the frequency of the water wavelength λ = 2 l (seiche). On the right edge of the tank near the vertical rigid wall R there is a weightless (compared to the weight of the displaced water) float, to which the electromagnetic force F ( t ) of the wave-generator is applied. The mechanical impedance of the electric drive of the wave maker is negligible compared to the impedance of the mass of water displaced by the float. The force F ( t ) and speed V ( t ) of the vertical displacement of the float are measured by appropriate sensors. The dispersion (frequency f [Hz] as a function f = Φ ( k ) of the wave number k [1/m]) of the propagating waves is shown in

There two experiments with pulsed excitation (

The wavemaker on the right wall R (

high-frequency waves at combination frequencies

The algorithms described in the article are constructed for the temporal representation of a boundary value problem. The presented algorithms are based on the use of high spatial-temporal resolution for fast switching wave regimes and don’t require the accumulation of information about wave fields and the boundary problem, using either only instantaneous field measurements or without them. The payment for smallness of information on the fields to be damped and the boundary problem in algorithm is high-frequency radiation. Above breaking algorithms cannot be reduced to either continuous representations (partial differential equations) or traditional discrete ones (point-like wise in space or (and) in time).

The author declares no conflicts of interest regarding the publication of this paper.

Arabadzhi, V. (2019) Four Algorithms for Boundary Control with Breaking in Space and Time. Journal of Applied Mathematics and Physics, 7, 2891-2901. https://doi.org/10.4236/jamp.2019.711198