A G-II FLC ALLOWS THEIR ACFT TO DSND BELOW THE ASSIGNED ALT WHICH PRODUCED A POTENTIAL CONFLICT WITH ANOTHER ACFT 100 MI W OF COU; MO.

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I WAS ASSIGNED AS CAPT OF GULFSTREAM X; A G-IIB; ENRTE FROM VNY TO COU. I WAS WELL RESTED; NOT HAVING FLOWN THIS; NOR ANY OTHER ACFT FOR 10 DAYS. THE DEP FROM VNY AND ENRTE TO A POINT 180 MI W OF COU WAS NORMAL. THE CTLR AT THAT POINT ASKED US WHEN WE WOULD LIKE TO COMMENCE OUR DSCNT INTO COU. HE WAS INFORMED THAT WE WOULD LIKE TO BEGIN OUR DSCNT BY REACHING 125 NM FROM COU. AFTER PROCEEDING BEYOND THE REQUEST POINT; WE REQUESTED DSCNT. WE WERE CLRED TO DSND AND MAINTAIN FL430. THE FO ENTERED THIS ALT IN THE ALT SELECTOR CONTROLLER (VNAV). I COMMENCED THE DSCNT. AT THE SAME TIME; THE FO LEFT THE RADIO TO OBTAIN ATIS INFO AND TO CONTACT THE FBO. WE WERE LEFT WITH LESS THAN DESIRABLE TIME FOR OUR DSCNT; AND RELYING ON THE AUTOPLT TO CAPTURE THE ASSIGNED ALT; MY ATTN WAS DIVERTED TO CALCULATING THE REQUIRED RATE OF DSCNT FOR THE REMAINDER OF THE DSCNT. WE DETECTED A TCASII TARGET LESS THAN 2000 FT BELOW US; CHKED OUR ALT WITH THE VNAV SETTING; AND REALIZED THAT THE CTLR AND AUTOPLT HAD FAILED TO CAPTURE THE FL430 ALT. AT FL418; WE REALIZED WHAT HAD OCCURRED AND COMMENCED AN IMMEDIATE CLB TO FL430. THE ACFT WAS HAND FLOWN FOR THE REMAINDER OF THE LEG. ON THE SUBSEQUENT LEG FROM COU TO CMH; THE VNAV WAS CLOSELY MONITORED; AND ONCE AGAIN FAILED TO CAPTURE THE PROGRAMMED ALT ON INITIAL DSCNT INTO CMH; AND WAS LEVELED OFF MANUALLY AT THE ASSIGNED ALT. THE VNAV WAS DISABLED AND PLACARDED IN ACCORDANCE WITH COMPANY MEL PROCS. THE TRIP WAS COMPLETED 3 DAYS LATER AND THE ACFT WAS RETURNED TO VNY FOR REPLACEMENT OF THE VNAV ALT SELECTOR. AS PIC; I WAS RESPONSIBLE FOR SAFE CONDUCT OF THE FLT. WHILE AT NO TIME WAS THERE IMMINENT DANGER DUE TO REDUCED SEPARATION; I SHOULD NOT HAVE TOTALLY RELIED ON THE AUTOPLT TO CAPTURE THE SELECTED ALT. MITIGATING FACTORS ON THIS WERE: ATC -- ATC'S FAILURE TO ANTICIPATE 'SLAM DUNK' DSCNT. THEY COULD HAVE AND SHOULD HAVE INITIATED THE DSCNT PRIOR TO THE 125 NM IF DSCNT WAS NOT POSSIBLE AT THAT POINT. EQUIP AND LOCATION OF EQUIP -- ALT SELECTOR FAILED TO CAPTURE PRESET ALT. ALT SELECTOR IS LOCATED ON CTR CONSOLE OUT OF EITHER PLT'S INST SCAN. RECENT EXPERIENCE -- THIS WAS MY FIRST TRIP AS CAPT ON THIS ACFT. MY ONLY OTHER RECENT EXPERIENCE WAS 5 LEGS AS COPLT; WITH NO L SEAT TIME AND NO TIME AT THE CTLS. I HAD NOT PREVIOUSLY OPERATED A GULFSTREAM II OR III FOR THE PREVIOUS 3 1/2 YRS. I DID RECENTLY COMPLETE G-II INITIAL TRAINING AND A SUCCESSFUL COMPETENCY; INST AND PART 135 CHKRIDE. COPLT DUTIES -- THE DELAYED DSCNT RUSHED THE FLC. THE COPLT'S ATTN WAS DIVERTED TO GATHERING ATIS INFO WHEN THE INCIDENT OCCURRED. SUPPLEMENTAL INFO FROM ACN 480414: WHEN TCASII ALERTED AND ATC QUESTIONED; WE STARTED A CLB IMMEDIATELY BACK TO FL430 AND HAND FLEW THE REST OF THE FLT. ONLY CORRECTION WOULD BE TO NOT RELY ON EQUIP AND NOT ALLOW CREW TO TAKE ATTN AWAY WHILE IN ANY DSCNT OR CLB.