Displaying similar documents to “On the representation of trajectories of bilinear systems and its applications”

Characteristic Cauchy problems and solutions of formal power series

Sunao Ouchi (1983)

Annales de l'institut Fourier

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Let L ( z , ∂ z ) = ( ∂ z 0 ) k - A ( z , ∂ z ) be a linear partial differential operator with holomorphic coefficients, where A ( z , ∂ z ) = ∑ j = 0 k - 1 A j ( z , ∂ z ' ) ( ∂ z 0 ) j , ord . A ( z , ∂ z ) = m > k and z = ( z 0 , z ' ) ∈ C n + 1 . We consider Cauchy problem with holomorphic data L ( z , ∂ z ) u ( z ) = f ( z ) , ( ∂ z 0 ) i u ( 0 , z ' ) = u ^ i ( z ' ) ( 0 ≤ i ≤ k - 1 ) . We can easily get a formal solution u ^ ( z ) = ∑ n = 0 ∞ u ^ n ( z ' ) ( z 0 ) n / n ! , bu in general it diverges. We show under some conditions that for any sector S with the opening less that a constant determined by L ( z , ∂ z ) , there is a function u S ( z ) holomorphic except on { z 0 = 0 } such that L ( z , ∂ z ) u S ( z ) = f ( z ) and u S ( z ) ∼ u ^ ( z ) as z 0 → 0 in S .